that is a really low budget.
first off, you have two options: to buy pre-built, or to build it yourself.
Building a PC is VERY easy. you read the 1-piece of paper instructions, you plug the thing in the only plug it can be plugged in, and you press the power button.
when you buy pc parts, each part comes with its own warranty. there will be no aftercare, but very few prebuilt pc sellers offer aftercare (i.e. "my windows doesn't change color" "my mouse cursor is the wrong size") and if they do, it tends to be very expensive.
Learning how to use a PC is something that everyone needs to do at some point .. unless they are willing to pay double and use Apple products.
You can build a PC easily. This is an activity you can do with your son and shouldn't take more than a few hours.
ok, second point. your budget is really low. at that price point, you will have to make hard choices that will be felt for some time. i recommend rising the budget a bit, because spending just a little more can get you much better modern quality components, and a better PC experience.
Also, a properly built pc will last you easily 4+ years without the need for an upgrade.
here is a reasonable part list, and why i chose each piece.
PCPartPicker part list:
https://uk.pcpartpicker.com/list/vDQjwV
Price breakdown by merchant:
https://uk.pcpartpicker.com/list/vDQjwV/by_merchant/
CPU: Intel - Core i3-8350K 4.0GHz Quad-Core Processor (£158.49 @ Aria PC)
Motherboard: MSI - Z370-A PRO ATX LGA1151 Motherboard (£92.15 @ Amazon UK)
this is a modern CPU. it's in the lower power bracket for the new Intel line, but it's cheap and *very* powerful. This is better than what i have in my gaming rig, and better than most professional machines in most businesses. It will easily run any single video card available today.
Memory: G.Skill - NT Series 8GB (2 x 4GB) DDR4-2400 Memory (£56.08 @ Amazon UK)
the standard RAM for any computer. no need to go any higher unless specific needs arise.
Storage: Patriot - Torch LE 480GB 2.5" Solid State Drive (£119.99 @ Amazon UK)
a reasonably sized SSD. a SSD is a new, modern version of the hard disk drive. there is NO comparison between a system that used a SSD and one that used a HDD, HDDs are slow beyond measure when compared to a SSD, and it has a real, tangible, very noticeable effect on everyday use when you have one or the other.
you could save £40 by buying a 256Gb but that would be a bit too small, instead 480GB is a very, very reasonable size.
Video Card: PNY - GeForce GTX 1050 Ti 4GB Video Card (£123.41 @ Amazon UK)
he wants to play, he MUST have a videocard. this card is the low end of the new generation, not great, but still reasonable. there are no real alternatives.
Case: Fractal Design - Core 2300 ATX Mid Tower Case (£43.20 @ Amazon UK)
yes, you can buy a £20 case and save, but here for $40 you get a case that isn't built out of spit and refuse. Fractal Design is one of the leading case manufacturers and here you are buying a product that is both very well designed, and that can easily last 2 or 3 generations of PC building.
cases are almost infinitely reusable.
Power Supply: Corsair - Builder 430W 80+ Bronze Certified ATX Power Supply (£42.73 @ Amazon UK)
corsair is a VERY dependable company with almost legendary customer support, and this is a adequately priced PSU that should easily last 5+ years.
Optical Drive: Lite-On - iHAS124-14 DVD/CD Writer (£11.17 @ Amazon UK)
you will need this, basically, to just install the operating system. DVDs are almost extinct. it does NOT read Bleuray, you need to buy a £40 model to do that.
Monitor: BenQ - GW2470H 23.8" 1920x1080 60Hz Monitor (£99.98 @ Amazon UK)
a 24 inch monitor (better and more modern and bigger than what i have) for £99 is a very, very good price. The brand is very reputable and the monitor itself should not leave anything to be desired.
Total: £747.20
You will also need: an operating system. This can be "bought" used (essentially you buy the product key) for less than the box price. I personally recommend Windows 7 as it is much, much easier to handle than Windows 10, but your choice is your own.
You will need a mouse and a keyboard, both of which can be had for £5 each, however a nice, £40 mouse is in order. Maybe wait until the next occasion to buy it.
I also suggest a mousemat, i personally recommend (with fervor) the Silent Monsters XXL
https://www.amazon.co.uk/Silent-Monsters-Gaming-Office-Mouse/dp/B01FFFSORA
which for £10 is absolutely AMAZING and way better than many more expensive models. i have yet to find a mousemat of better quality.