For simple games like that . . well, I have an ASRock A300W DeskMini running an old Athlon 200GE, but the next generation of it is the X300W, which supports the 5600G and 5700G.
The W version comes with wi-fi.
https://www.asrock.com/nettop/AMD/DeskMini X300 Series/index.asp
Get some laptop RAM (yes, these are desktop PCs, but they use laptop RAM), storage, and a CPU, and you're good to go.
Note as well, that there are also low-power variants of those two CPUs. The 5600G and 5700G are rated at 65W, whereas the 5600GE and 5700GE are rated at 35W. The GE versions have a lower base clock speed, but the boost speed is the same. Though, I think
their performance is a little less than the G versions because they proably thermal-throttle more quickly to stay in the 35W envelope.
I'm also given to understand that you can use the Wraith Stealth cooler that comes with the 5600/5700 in these units rather than the small one that comes with the X300, but you have to remove the shroud at the top of the Stealth cooler. I haven't tried it myself, so, don't hold me to that. I've seen posts online with the A300W that it works, though.