Yeah £300 is not a lot sorry, you could buy something like mentioned - though even a 3200G build is going to push that budget. It's a good little CPU but a single 8GB stick will hamper it even more - some memory is assigned to the built in graphics and single stick performance on Ryzen isn't exactly stellar to start with.
ULTRA FAST CCL 4.0GHz AMD 3200G AMD Ryzen 3 Gaming PC - 16GB RAM, 1TB HDD | eBay
- would be slightly better - twin RAM sticks, 1TB disk (though not an SSD) but reasonably you would eat up 240GB in literally no time at all.
Now, just to throw something else in the mix - if your internet is decent enough (and you have an old laptop or something, they do a mini box thing too if they are still doing them) you could opt for something like Shadow cloud PC - Shadow - Tech specs - Check components CPU, memory, storage, GPU
- £12 a month gets you a PC far better than you could buy for £150 or thereabouts a year. Albeit it is not technically yours to own, but you would lose that in a year on parts you buy today realistically if you sold them off next November
I used the service myself for about a year, it's a decent option and you can bring along your own games to it - not tied in like the retched Stadia service.
The other option is Geforce Now which is similar, £5 a month (at the mo) but has a more limited selection of games, though it is a long list and is JUST games unlike Shadow which is basically a PC so you can browse the web on it or do some office work or 3d design etc.
Literally a cheap laptop or desktop plus one of the above can get you gaming