Any phones below $200 capable of running light games?

[Ultimatesaber27]

I finally bought a PS4 controller (though I was a bit unwilling to do so). Connection over Bluetooth is horrible, I can't play any Mega Man game like this, even on Easy mode. Tried to use the DS4 wired, but when I connect it to the controller, all it does is charge (or that's what I assume the yellow light to mean).

Any way to make the DS4 work wired?