Lighting – Halogen light stops working when cover is installed

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 first linked site appears to concern itself wholly with halogen bulbs. It claims of low-voltage halogen bulbs that, "They also produce a nicer light, with warmer, brighter and more vibrant colors."

The second site's entire argument for low-voltage light being better seems to rest entirely on how configurable and remote-controllable low-voltage lights are, as well as how much easier it is to scatter gobs of them about to achieve your lighting plans for world domination.

I am inclined not to trust the first site, because their argument doesn't make intuitive sense and they provide no citations to back it up. The second site's argument makes sense on its face.

I can't speak to halogen bulbs, but when it comes to LEDs, I would not expect any difference in light quality between a line-voltage and low-voltage LED.

I don't know of any LED bulbs that actually drive the LEDs at line voltage. The ones that you can just screw into a line voltage socket just include their own DC rectifier, the same way that CFL bulbs include their own ballast to drive the fluorescence.

I would expect you can get finer control by putting your own transformer in front of a line of low-voltage LEDs, and your resulting lighting system will probably be more efficient and flexible. (As an example, I know an LED retrofit lamp I recently installed could dim down to only 5%, whereas you might be able to dim the LEDs throughout their full range using your own transformer.)