I'd +1 the desktop idea unless you really need the portability of a laptop. They're easily serviceable, you can put it together with exactly what you want, and upgrading later is often really cheap with second hand parts. They're also easily cleaned - laptops much less so, especially if you're going to be running games on it (more load = higher fan speed = more airflow = more dust buildup.)
My decade old desktop is still going strong as a media PC after whacking an SSD and second hand upgraded CPU in it - that's the only upgrade it's had from new; cost me just north of £60 for both parts (and it was a budget machine when it was put together!)
"Gaming laptops" (basically meaning they have dedicated graphics) will also usually blow your £500 budget out the water.
That said, whatever machine you're after (laptop or desktop):
- Don't accept anything less than an SSD these days, ignore "HDD" or "hybrid" drives off the bat - and I'd be looking at a 256GB drive;
- Preferably 8GB RAM, minimum 4GB;
- Look out for an optical drive if you still need it - many machines these days come without;
- CPU spec is generally less important than people tend to think these days, but with a budget of £500 I'd be shooting for an i5.
- If battery life is important to you, check it before buying (many forget this bit!)
Ebuyer is a good place to look generally for these things - they're often among, if not the cheapest and I've never had any issues with them.
Quickly pulling one out that's around your budget and meets the above specs:
Lenovo V130 Laptop - Laptops at ebuyer - https://www.ebuyer.com/874325-lenovo-v130-laptop-81hn00nguk