Are you sure the computer is using an ATX compliant PSU?
HP uses ATX connector with a custom wiring not compatible with ATX on some machines, you can find ATX pinout on the Internet and compare the color of the wires to your PC connector. If that is matching, you can replace the PSU. Nothing can fry but if something requires too much currant, the PC will just stop. But I highly doubt a stock Pavilion will drain 20A or more on +5V...
The Radeon HD 5570 has a TDP of 39 W and is powered through the PCIe on the +12V line I suppose. That is 17% of the max +12V can output of you current PSU.
What is the computer model?

All the wires match the standard. The only two things worth noting are that pin 13 has two wires - orange and brown. From what I've found, the brown wire is a voltage sensor. Do new PSUs have this? Does it matter?

Pin 22 has two wires - red and pink. Wikipedia indicates it's a +5 V sense wire, but it's a non-standard practice and not part of the official ATX standard. Does it matter if the new PSU doesn't have this?

The computer is a HPE 410t. The motherboard might be the HP Iona-GL8E.

I was trying to calculate how much power the DVD drives use. Would I be able to copy a CD/DVD with the DVD writer drive? I could still use both drives at the same time with a new PSU?

Two HDD would be 13 to 18 W total.
Two SATA DVD drives would be 30 to 54 W total.
So that's 72W or 14.4A, I think.
I'm not sure what else uses the +5V rail. USB does, right? But with 8 USB ports, it can only draw up to 2.5W each which is 20W or 4A, correct? So that's a total of 92W or 18.4A. That's assuming everything is maxed out. So if my math is correct, a PSU with 20A on the +5V rail, should be enough, correct?

Maybe have a look into this thread:

Any modern psu's that have large 5v rails

There are some recommendations for newer PSUs in older PCs.

What I have found is your PSU is a HP 585007-001 (LiteOn PS-5301-08HF) or HP 585008-001 (LiteOn PS-5301-08HA) but was not able to find the said sensing lines...
I don't how much reliable The Retro Web is, but have a look here https://theretroweb.com/motherboards/s/msi-ms-7613-ver-1-0.
It is written ATX1 in the mother and the web site state it is ATX 2.x compliant.

Respectable PSU manufacturers only uses one wire for each pin. If the standard define the use of multiple identical lines it is to split the current between this lines.
So the possibilities here is that your sensing lines are only for the PSU to detect potential issues and have no use for the motherboard. BUT I have seen Socket 370 HP Pavilion with ASUS manufactured motherboards refusing to start with anything other than the original PSU... So can be possible here too...
But for me, if the ATX connector of the original PSU is correctly wired, that mean the motherboard is ATX "compliant" (at least on the voltages) and there should be no problem testing with any ATX PSU to see if it work. Logically, nothing can happen to the motherboard as it find what voltage/ground it needs where it need it. It can just not start (because of weird proprietary nonsense in the original PSU maybe because of the gray or purple lines) or start then stop randomly because there is not enough current delivered (I highly doubt that though). And any modern good quality PSU have enough security embedded to detect shorts, power surge, etc and stop by themselves before anything can fry anyway.
And if the motherboard refuse to start with another PSU, you can try to find a more powerful compatible one. For example the 460W HP 570857-001 is the one sold with the HP HPE-410FR models and should fit fine.

That said, it is up to you to take the risk to test it and I cannot be held responsible for any damage that can occur. Hope I have been of some help, I cannot do more.

For both HP and Dell: if the motherboard is new enough to have PCI-E, it's pretty much guaranteed the ATX connector is standard pinout. At least I haven't see one that isn't so far, and I've worked on quite a few OEM machines back in the mid-late 2000's.

That being said, the motherboard is new enough to power its CPU from the 12V rail via 4-pin 12V connector. So the 5V rating (and thus combined 5V+3.3V rating) is irrelevant. A CD and DVD drive will rarely use more than 1 Amp on the 5V rail (most of the time, it would be much less, even when the drive is in use.) Same goes for mechanical HDDs. The chipset(s) and RAM are going to be the biggest consumers on the 5V and 3.3V rails, depending on how the motherboard is designed and which chipset / Northbridge + Southbridge it has. Generally, though, these won't consume more than 20-40 Watts combined... so again, the 3.3V and 5V rail ratings are pretty much irrelevant with newer machines like this.

When it comes to replacement PSUs, just pick one that will work with your current and future needs. This will particularly depend on what graphics card / GPU you want to use with the system, as that will likely be the biggest power consumer if you end up going with a high(er)-end one. Otherwise if you stick with the current HD5570 you have, a good quality 250-300 Watt power supply will be more than enough. In fact, even with slightly more power-hungry mid-range GPUs, you would still be fine to use a quality 250-300 Watt PSU. Whatever you do, don't go on Amazon or Ebay and buy some no-name 1000 Gigawatt PSU that's lighter than a happy mean, even if it has supposedly "good" reviews. I personally like to stick with OEM PSUs, as they are built rather well. Of course, some of them do suffer from bad caps (and if I had to guess why your current PSU is dead / not working, that's very likely the issue with it.) But overall, I still very much prefer these OEM units to no-name new garbage. The OEM PSUs are also plentiful and cheap, so you can get 2-3 for the price of a much more powerful name-brand unit (actually, many of the "branded" PSUs use the same companies as the OEM PSUs, so there's not that much of a difference - i.e. Delta, LiteOn, Chicony/HiPro, FSP/Fortron, AcBel, and Channell Well, to name a few.) The alternative is to buy a decent/good "branded" PSU at a higher price and not have to worry about it, at least for a while (i.e. Seasonic, Antec, a good deal of Corsair and EVGA units, and a few Thermatake units.... among others.) As for the old PSU - don't throw it away if you have no use for it. Either sell it or give it away locally as some people (like myself) do occasionally buy broken units off of eBay to repair for fun or use as parts. Almost all of my desktops run either repaired or salvaged PSUs I got for free or cheap.

momaka wrote on 2024-03-19, 13:41:

For both HP and Dell: if the motherboard is new enough to have PCI-E, it's pretty much guaranteed the ATX connector is standard pinout. At least I haven't see one that isn't so far, and I've worked on quite a few OEM machines back in the mid-late 2000's.

That being said, the motherboard is new enough to power its CPU from the 12V rail via 4-pin 12V connector. So the 5V rating (and thus combined 5V+3.3V rating) is irrelevant. A CD and DVD drive will rarely use more than 1 Amp on the 5V rail (most of the time, it would be much less, even when the drive is in use.) Same goes for mechanical HDDs. The chipset(s) and RAM are going to be the biggest consumers on the 5V and 3.3V rails, depending on how the motherboard is designed and which chipset / Northbridge + Southbridge it has. Generally, though, these won't consume more than 20-40 Watts combined... so again, the 3.3V and 5V rail ratings are pretty much irrelevant with newer machines like this.

When it comes to replacement PSUs, just pick one that will work with your current and future needs. This will particularly depend on what graphics card / GPU you want to use with the system, as that will likely be the biggest power consumer if you end up going with a high(er)-end one. Otherwise if you stick with the current HD5570 you have, a good quality 250-300 Watt power supply will be more than enough. In fact, even with slightly more power-hungry mid-range GPUs, you would still be fine to use a quality 250-300 Watt PSU. Whatever you do, don't go on Amazon or Ebay and buy some no-name 1000 Gigawatt PSU that's lighter than a happy mean, even if it has supposedly "good" reviews. I personally like to stick with OEM PSUs, as they are built rather well. Of course, some of them do suffer from bad caps (and if I had to guess why your current PSU is dead / not working, that's very likely the issue with it.) But overall, I still very much prefer these OEM units to no-name new garbage. The OEM PSUs are also plentiful and cheap, so you can get 2-3 for the price of a much more powerful name-brand unit (actually, many of the "branded" PSUs use the same companies as the OEM PSUs, so there's not that much of a difference - i.e. Delta, LiteOn, Chicony/HiPro, FSP/Fortron, AcBel, and Channell Well, to name a few.) The alternative is to buy a decent/good "branded" PSU at a higher price and not have to worry about it, at least for a while (i.e. Seasonic, Antec, a good deal of Corsair and EVGA units, and a few Thermatake units.... among others.) As for the old PSU - don't throw it away if you have no use for it. Either sell it or give it away locally as some people (like myself) do occasionally buy broken units off of eBay to repair for fun or use as parts. Almost all of my desktops run either repaired or salvaged PSUs I got for free or cheap.

You are wrong, for example HP Zxxx series for each gen uses PCIe motherboards and each time they use a different and non-standard connectors. And I also have a HP Xeon tower (don't remember the model, but not a Zxxx) that have a PCIe motherboard that is using the ATX 2.01 connector, but wired differently... Unofficial adapters exists that allow to use a standard ATX PSU though.
With HP (and apparently Dell, but I know almost nothing about Dell hardware as I try to avoid it) nothing is simple. On one side they are part of different consortium to create standards and on the other side they do not use one when it exists. Of course that is to force the customers (mainly businesses) to use the customer service to buy their official parts and services. They sell products they want to control. They are not the only ones. It is disrespectful to the customer but legal. Knowing that a customer have to be careful when working with such hardware and check everything to not make fatal mistake.

Sorry, perhaps I should have been a little more specific: what I meant to say applies mostly to regular desktop (tower, SFF, and USFF) machines. Workstations have always been a little different from both and indeed rarely used standard ATX pinout (be it because of dual and/or high-power CPU support, or just for the sake of it.) Also, my knowledge of Dell and HP stuff dates back to mostly socket 478, 775, and early 115x era, so what I mentioned might be a bit outdated now.
Still, a quick search of the O/P's motherboard on google revealed a pretty standard-looking mATX form factor and I'm pretty certain the ATX pinout on the mobo is standard too. At least for a while after the s478 era, both Dell and HP made relatively standard stuff. I think around the era of 4th gen Intel i-series is where the regular desktop (tower, SFF, and USFF) machines really started to get "quirky" again... though this time mostly in the form of non-standard form factor PSUs, particularly for the SFF and USFF systems (and many of these don't even use a 24-pin ATX connector altogether.) I haven't been following too closely what happened after this era, as I strayed away from IT and PC hardware for a while. Didn't come back to system building until late Intel 6th gen / early 7th gen i-series again. And at that point, it was mostly SFF and USFF machines running either external or those non-ATX form-factor PSUs (but most dominant were sales of AIO machines and laptops.)

lilblu wrote on 2024-03-19, 03:14:

All the wires match the standard. The only two things worth noting are that pin 13 has two wires - orange and brown. From what I've found, the brown wire is a voltage sensor. Do new PSUs have this? Does it matter?

Some PSUs might have it, some might not. For PSUs that have it, it is indeed a feedback voltage sensor (for the 3.3V rail). It is irrelevant to the motherboard, though. All the motherboard cares is that it gets good stable 3.3V.
The reason why some PSUs have it (especially older ones from back in the day when heavier 3.3V and 5V rail loads were the norm) is to detect the voltage drop of the 3.3V rail over the ATX cable and try to correct for this by sending a slightly higher voltage from the PSU so that when it reaches the ATX connector on the motherboard, it is exactly 3.3V.

lilblu wrote on 2024-03-19, 03:14:

Pin 22 has two wires - red and pink. Wikipedia indicates it's a +5 V sense wire, but it's a non-standard practice and not part of the official ATX standard. Does it matter if the new PSU doesn't have this?

It's the same thing as with the 3.3V sense wire. Only matters to the PSU. Motherboard doesn't care.

No, your new/replacement PSU does not need to have these sense wires.