Washing is not likely to help if you've actually gouged the plastic.
More expensive players do not necessarily skip less (and of course, it would be much cheaper to simply buy a new CD, assuming it's still in print). But yes, you might have luck reading it on your computer, that's easy enough to try.
Otherwise, you can try polishing out the scratch. You need a very gentle rubbing compound - best bet is to just buy a cheap 'CD Repair Kit' at your local music store, which should have a small tube of the stuff.