The simplest solution is to replace the perspex. I have a plastic polishing kit which contains superfine wet n dry, starting at 1500 grit and going in stages to uberfine x0000 grit, plus liquid polish. It works but is very tedious and time consuming and is not meant for whole areas - just localised refinishing.
http://www.frost.co.uk/plastic-polishing-kit.htmlhttp://www.glasspolishshop.com/plastic_polish 
