The stuff deigned for this sort of problem is Greygate Plastic Polish. Used it for years on visors and any other plaggy/ acrylic surfaces that have fine scratches and other blemishes. Brilliant stuff (it's used in the Aero industry for removing scratches from aircraft canopies).
Cheapest I've found on eBay is £7.50 delivered, the 100ml bottle should last you yonks as you only need a gnat's chuff each 'rubbing'...
I managed to spray my screen with petrol when i first got the bike and was left with some fairly large stain marks on it to the extent i was looking for a replacement screen. As an experiment i used Yoshimoto Deep Gloss polish i had in the shed on it which has completely got rid of the huge stain
Costs about a fiver for a 250ml bottle and i use it on the screen all the time now to keep up the finish