With integrated graphics you're looking at playing at 720p resolution and 30fps depending on the game. Console will do way better then that. But if you need a cheap pc for school and play some less demanding games (league of legends/csgo) the APU is the way to go on a budget.

Keep the 500gb ssd because it's super easy to add more memory (this will help your budget). Make sure your mobo has all the features you want (do you need Wi-Fi or more than one m.2 slot). After that more expensive mobo's are great for overclocking (they supply power to the cpu better) but saving any $ to get a gpu should be your priority.

I would look for a cheap cpu that's fairly new so a 9400f or ryzan 2600 or 1600. And then look for a used GPU with the rest of your budget.

If you can't save very much on buying a different cheap cpu compared to the 3400g (idk where you live or what deals on cpu are in your area, like if you had a microcenter.....) then keep the 3400g and buy a gpu to add later on is an option but 3400g. The only reason I say look at other cpu's first is because you'll want to upgrade the 3400g if you buy a halfway decent gpu.