I'm pretty sure that these plastic headlight housings have a factory applied (sprayed) clear coat as most vehicles do.
Toothpaste and many other methods on Youtube do not work very well and the results don't last very long.
On my 98 Dodge minivan I followed the method I saw on Youtube posted by a guy who does restoration work exclusively on very expensive and exotic cars, Ferraris, Lambos etc. = Sand the clearcoat off using several grits of sand paper down to a fine grit so they are nice and smooth. They will be completely clouded at that point. THEN mask off the lights and apply several light coats of GLOSS clear coat paint. Automotive gloss clearcoat would be best but I used Rust-O-leum gloss clear coat from Walmart. There is a Gloss Clear Coat 2X Ultra Cover and if you use that one spray it on very quick and light coats because twice as much paint comes out of those nozzles. Too much paint and it will run but if you have a rag you can quickly wipe off the run and re-coat it. Such mistakes cover up quite well and will be very difficult to notice. The completely fogged lights from the sanding came out crystal clear, like magic, like brand new. Perfectly restored and will probably last another 20 years
According to that auto restorer of very expensive cars, $250,000 and up, in NYC or New Jersey, that is the correct way to restore headlight lenses. It's also quite easy and inexpensive. That's the way I am going to do my truck headlights when I can get around to it. Would be a good job for this weekend.