Well you're going to end up with something with a relatively low-end CPU. Maybe an intel i3, or i5 if you are lucky. If a lot of storage space isn't needed, I would definitely recommend something with an SSD drive, as that will make the desktop experience much faster.
But this is used too, although for under £350 you'll be looking at OK new stuff or great refurbished gear, mainly in the red corner with AMD at your price point.