It's a recessed (canned) light cover (but I'm guessing you probably already knew that).
As "ppumkin" suggests in above comment, try to see if you can find a manufactures name (& model number) on the actual light fixture. Having such information to hand will greatly help you in tracking down an appropriate replacement cover.
Also you might want to check the bulb (Wattage & type) that is currently in there, as it really shouldn't have done such damage to the light cover.
Nowadays you can get compact fluorescent bulbs specifically for recessed lighting, or if money isn't an issue, look into available LED lights for use in recessed lighting.
The answer is not a cut and dry one in your case. The troubleshooting is a process. You will need an understanding of a multiple light parallel circuit. You will also need a proximity type voltage tester and possibly a VOM.
Assuming the voltage feed starts at the switch (not always the case, but normal) you will need to verify input voltage at the line side of the switch with the switch in the off position. If you have voltage there, turn the switch on and verify voltage on the load side of the switch. At this point, a VOM is handy to test voltage across the hot and neutral and/or ground. If this looks good, proceed to the closest light fixture, and with the switch on, test the center hot tab in each fixture with the no-touch tester. In your case I would think you may not see any voltage based on your question. This could mean that you have an open neutral, an open hot, or at worse, a shorted hot to neutral/ground.
Since there are several possible reasons for your condition, it would take a lot of tutorial to explain every possible scenario. Assuming you do not see voltage at the fixtures, the basic technique for troubleshooting will be to start at the last known verified voltage point, then follow the wiring and check all connections in the junction boxes. Visually check the bulb sockets for broken or shorted metal tabs. This should be done with the power off at first, looking for obvious loose or disconnected wires, then with the power on using your voltage tester. Since this condition occurred after you adjusted the height of the sockets in the fixtures, I suspect tension on some wire may have pulled a wire out of a socket base or out of a wirenut in the j-box. Unfortunately, most can/pot lights have a built in j-box attached to the top of the fixture. This necessitates dropping the fixture down below the ceiling to access the j-box, or getting access from above. (attic).
Basically, you are following the circuit looking for an open or short, just like following a hose, looking for a water leak or stoppage.
This is not a hard job, but extreme caution must be taken when testing energized circuits. If you do not have good electrical skills, the proper test equipment, or a logical understanding of switched paralleled circuits, then this job is better left to a pro.
Maybe one of my buddies here on SE can add a good graphic showing this type of circuit and the test points. A simple line drawing showing the switch and junction points would be a great edit.
Best Answer
Its been painted, and badly at that, which is going to make everything stick together. Get a utility knife and score around the edge of the flashing, a.k.a the baffle, and see if it will come off after that. Modern day recessed light baffles are held in place by three metal tabs that push on the side of the can. They are like springs. When I take my down to replace the bulb it feels like I"m going to pull the whole thing out of the ceiling. The thing has a death grip on the can. Could be similar with yours, in which case the flashing will pull straight down. On my cans I had to use a metal L shaped dental pick to get under the lip of the can so I could get my fingers under it and start pulling.