My starter sometimes crank the engine without problem and sometimes only produces a loud click without any thing appearing to turn. The battery is brand new

I cleaned the battery terminals, the lead wire to the starter and the two connections where the negative cables are connected respectively to the engine and the body. I have also connected a jumper I feel like I am playing with a wheel of fortune the fault is not with the ignition switch because
if I try to start the engine by turning the key or by hot wiring the starter itself, the result is no different. If the car does not start it helps to turn the engine by hand with a wrench or by turning the power steering pump pulley.and try again with the key.

I am starting to wonder if the toothed wheel(s)on the starter or the flywheel could be a problem or maybe the solenoid in the starter itself.

What are the thoughts out there