A readily available volatile solvent (lawn mower and leaf blower fuel).
Similar to gasoline, I stumbled onto carb cleaner as a restorer. A few drops got onto the old cloudy Marquis lens, and dried really clear. Since then, though, I usually just wet-sand down to 800 grit, let them dry thoroughly, and then use a fine 1" wide artist’s brush to finish with acetone. The solvent dissolves the surface of the plastic, letting the scratches and cloudiness flow out (to borrow a car painter’s term). The surface of the plastic does remain tacky for a while, so don’t touch it until it’s hard again.
I have refinished a ton of headlights, and here’s what I use for a spray on clear coat:
Rustoleum Clear, UV resistant, adheres to plastics
Here’s a fantastic how to. Of the methods that I’ve used, it gives the absolute best results: