For 600 quid, you can get a build done, but it will be without dedicated GPU. Since as said above, GPU prices are currently through the roof due to the short supply.
Here's one option:
PCPartPicker Part List
CPU: AMD Ryzen 3 3200G 3.6 GHz Quad-Core Processor (£178.00 @ Amazon UK)
Motherboard: Asus PRIME B450M-A Micro ATX AM4 Motherboard (£66.78 @ Aria PC)
Memory: Patriot Viper 4 Blackout 16 GB (2 x 8 GB) DDR4-3000 CL16 Memory (£74.09 @ Ebuyer)
Storage: Samsung 860 Evo 1 TB 2.5" Solid State Drive (£91.98 @ Aria PC)
Case: Cooler Master MasterBox Q300L MicroATX Mini Tower Case (£44.92 @ More Computers)
Power Supply: SeaSonic FOCUS 550 W 80+ Gold Certified Fully Modular ATX Power Supply (£77.19 @ Amazon UK)
Total: £532.96
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2021-05-08 18:27 BST+0100
Few words.
Put in AMD APU, R3 3200G, since it has Vega 8 built-in graphics that can be used for low-end gaming. R5 3400G is better CPU with better, Vega 11 graphics but it blows your budget. And it also needs dedicated CPU cooler. While most Intel CPUs also come with built-in graphics, AMD's built-in graphics are more powerful than Intel counterparts.
Mobo comes from Asus, micro-ATX in size and with 4 RAM slots, for future RAM expansion.
Note: Only issue with this build, is that your MoBo needs to have latest BIOS for it to work with the CPU. If MoBo doesn't come with latest BIOS, bring your PC to local PC repair shop and let them to update the BIOS for you, for a small fee.
Also put in 2x 8GB RAM, at 3000 Mhz. Since this is AMD build, your RAM can run 2933 Mhz at most (due to AMD chipset).
For storage, went with 1TB SSD, Samsung 860 Evo, which is solid SSD to use (my Haswell build also has 860 Evo as OS drive, full specs with pics in my sig) and 1TB is more than enough for OS and your games.
PC case is personal choice and feel free to switch it out. But for time being, i put in a nice and customizable cube case from Cooler Master. PC case has plenty of ventilation, with magnetic filters.
specs:
https://www.coolermaster.com/catalog/cases/mini-tower/masterbox-q300l/
And lastly, picked a good quality PSU for you: Seasonic Focus+ 550 (80+ Gold). PSU is fully modular, has good efficiency and comes with 10 years of warranty. PSU is also powerful enough to power any future dedicated GPU, as long as the GPU TDP doesn't exceed 200W. (All 3x of my PCs are also powered by Seasonic, full specs with pics in my sig.)
All-in-all, this build is good to get you going, where you can game on. And it has plenty of future upgrade options. E.g when you collect more money and/or GPU prizes normalize in 2022, you can buy a dedicated GPU and greatly improve your gaming performance.