I have been using a small 6" palm orbital polisher and the cheap wal-mart toothpaste for YEARS.
Wash the headlights of any oil/dust/road grime.
Use masking tape to mask off around the lights. Don't want to scratch the paint!
Put a 3 inch line of toothpaste on the damp polisher and place on headlight. THEN turn on.
I use a large bucket of warm water and a cup to keep things wet and lubed.
You will see the toothpaste "foam" turn yellow as it pulls out the oxidation and gunk as you move it around the headlight lens.
I suggest a chair to sit on as you polish.
Rinse and repeat with fresh toothpaste often.
This works on those SEVERELY yellow lights on Dodge and Ford cars from the 90s.