I did pottery for fun (not profit) for a year or so several years ago. I, too, never quite got the hang of centering, nor wheel work in general, but I think I know what you’re talking about.
It may be that you are using clay with very heavy “grain” (?) I don’t remember the proper term for it, but usually beginners work with clay with bigger granules in it because it tends to be cheaper and more durable. This is opposed to something like porcelain, which is super-smooth and beautiful to touch, but is more difficult to control.
So, with the heavy granules, plus the unnecessary pushing on the wheel - I used to do this too, by the way - and perhaps not enough water, all combine to create the gravelly friction against your hand. Ow!
My suggestion: brace your hands/arms as instructed, but instead of pushing your right hand into the wheel, push up with your right hand a bit against your left, so you’re offering yourself some resistance, and it will keep your right hand just above the spinning wheel. It also can be a bit of a nice resistance workout.
As for adding more water, it’s probably a good idea. I tended not to use enough either. But do be careful, as I think TOO much water might make the piece slippery on the wheel.
Hope that helps. Good luck! Reading this made me miss potting.