Damage in the center is caused by the drive spindle, or more precisely by the spring-loaded clamp that catches the disk to rotate it. That's fine, it'll work as long as there is no actual damage like holes and tears, although it might slip in some drives if enough coating is removed. These are cheap floppies. Proper ones had an extra ring here from sligtly thicker and sturdier material for the drive to grab onto. But alas by the time 1.2MB floppies got popular the requirement of cheap mass manufacturing demanded cost cuts.

Scratches visible in the head window are bad. And no, you can't fix it, trying will only make it worse (most likely damage the floppy permanently).
If you care about the floppy itself, make a copy and use that rather than the original. It's the only way. There is mechanical contact between the head and the disk (this is not a closed, dust-free HDD spinning so fast that heads can rest on a air gap cushion) and the material will be slowly eroded away. Any dust that gets in there will accelerate the process.

Yep I've got that problem too with some disks. However make sure your drive's head is clean as it might damage some disks.

Also when a floppy do this when the heads are clean, be aware that the coating that comes off the disk will get stuck on the heads, so after using this kind of bad disk, your drive's head might have become dirty. And you know what's worse ? Like I mentioned a dirty head can damage perfectly fine disks, so this might "spread" the disease to good disks and break them too.

So yeah when you see disks with ring scratches in the window, make sure your drive's clean every time.