Back to your original question - Yes, you can remove it and repair it (depending on what was broken). The cover itself is kinda like a ziplock bag. There is a 'ziplock' fastener at the bottom of the headrest. You don't normally see the fastener, because the cover is designed to hide the 'ziplock' from view. Anyway, if you remove the headrest, and turn it upside down so you can see the bottom of it, and then if you will pull on the fabric, you'll see what I'm talking about. The ziplock just pops into place.
Once you get the ziplock open, the cover can be easily removed. On the inside, you will probably find that you have stripped a bolt out of a plastic thingamajig. A larger or longer bolt will probably cure the problem.
