Wash the scratched area with soapy water. Liquid dish soap will work well for this. Be sure to wash the scratched area as well as the area surrounding it.
Take a piece of 400 grit wet/dry sandpaper and wet it. Sand the scratched area in a circular motion until it looks cloudy. Make sure the cloudiness is even all around.
Move up to the finer 1500 grit wet/dry sandpaper. Get it wet and sand the same area. The scratches produced from this will be much finer. Continue sanding until it looks even.
Apply a rubbing compound to the scratches. Buff the area with a foam pad at low speed. Continue buffing until the Perspex is clear. If it is still not as clear as you would like it to be, you can use a finer rubbing compound.
Rinse the Perspex off with water. Dry it with a clean towel.