This is about as low as I would go and be playable @1080p medium or better settings in every game out. This includes the OS "Fully legal version by Microsoft's resell policy" You would only need to use another PC to make a Bootable USB flash drive to install the OS. Sales and rebates that come and go would drop the price $50.
I forgot to say my budget right now is £550 but I can go up to £600 if necessary
Swap out the g4560 for a i5 7500 and upgrade to a RX 470. You could also run a Ryzen 5 1500x for about the same price as the i5. Both option will push you into the 570 range.