What should I check first when a pool pump hums but won't push water?

It’s almost always an air leak. Check the skimmer basket and pump basket for debris, then inspect the pump lid O-ring — a cracked or misaligned one sucks air. Lube it with silicone grease and reseat the lid. Also make sure the water level in the pool is high enough to feed the skimmer. If that doesn’t work, look for a tiny suction-side leak using shaving cream around joints.