The only time I have done them I used J-Wax Kit. Now called Formula 1
http://www.formula1wax.com/products/car-wax/carnuba-paste-wax
No particular reason, it was what I had laying around. Not only does it have wax and some solvents and cleaners, it has a real fine abrasive that does not leave scratches.
Plus, the wax may slow down the UV.
I put some on a wet lambswool polishing pad on a drill. Kept it wet to keep it cool and not melt the plastic. Also kept the drill turning slow and not much pressure. Actually, the very first time on a headlight, I used some Kit on a rag. Little rubbing and I had a clear spot. Then I went to the drill.
About 35 years before that, back when headlights were glass, I had several Coke machines that had a problem with the buttons. Unlike all the other makes and models, these had buttons that were about 2 inches by 2 inches, and were at least an inch thick clear plastic.
The label went in a slot in the back side, and you read it through the plastic.
Trouble was, that model was not made anymore, and you could not get new buttons. Surprisingly, the buttons had not clouded up despite being in the sun for years.
Instead, they had "crazed", developed hundreds of tiny cracks. So many, that in many cases you could not read the label.
I was taking some out of a machine we were going to scrap, and was sorting them as to the best of the bunch. When I looked in from the side, I realized that the cracks were all confined to the top 1/16 or less of the button.
They did not go deep at all.
I took one of the worst ones, and used a coarse grinder to grind off the cracked part. This left me with a completely opaque button. Then I used a finishing sander to smooth off the surface and clean up the grooves and scratches the grinder left.
Then I hit it with some polishing compound I had laying in the shop. Had no Kit, probably would have been too wimpy anyway. When I was through, I had a perfectly clear button.
I refined my technique with the next one by using the finishing sander first. Little slower to get past the cracks, but overall faster. When it was all said and done, I had done about 60 buttons or more. Put over 10 machines back in operation.
Don't recommend anything that drastic for headlights though.....