I have a 175' run from panel to an outlet for a swimming pool pump that says 10 amp on it. What size cable do I need to use to make that run.

Assuming 240 volts, which is a value needed to do voltage drop calculations, I would use #10 copper wire.

One problem with your plan, cable is not allowed for pool wiring.

You can use 3/4" PVC conduit, buried 18 inch deep, and pull three #10 THHN/THWN conductors.

There are other requirements for pool wiring, ground fault protection of the pool pump circuit, 120 volt outlet required nearby, grounding, etc. You need to consult with your local inspector to learn what their requirements are for these issues.