I think there's a few options you can do here depending on funding and what you can be bothered to do. From experience those little Aiwa systems always seem to have a half decent amp in them, for what they are anyway, so maybe look to upgrade your speakers first. You can get some pretty good speakers for <£200, even for <£100, or even second hand which would be a major improvement to your current speakers. There's a few speakers available which do very well at improving those little mini systems and at the same time provide you with access to the first run of the hi-fi ladder , if that's the way you fancy going.
Better speakers would give you a better sound straight away and then allow you to maybe upgrade the amp at a later stage when funds permit.
You could splash out and replace both your speakers and amp - you'd be surprised what quality you can get these days for not much money at all. There's plenty people round here that would help you buy the best you can for your money.
Another option would be to do away with both the speakers and the Aiwa system and get yourself a more traditional PC sort of system. As well as others, Creative do some good speakers for around the £50-£60 mark which are perfect for gaming and general music listening, we're not talking hi-fi quality here though, definitely not the type of sound you'd expect from a dedicated amp and speakers of course but definitely good enough for the general sort of PC usage it looks like you do. The more you pay with regards to PC speakers the more you get but there's a good set of speakers to match whatever budget you have.
As a little free tip for the time being though, get yourself some Blutac and put a dollop under each corner of your speakers. It'll isolate the speakers from your desk, reducing vibration, and help to give a better sound in general. Also, I don't know those speakers but do they have a rear sound port in them? If so they look like they're too close to the wall and will benefit from a gap of a few inches.
I'm sure someone else more knowledgeable will post with better ideas but hopefully that's given you something to think about, and maybe somewhere to start.
Good luck
