The answer to 'why is my P-trap won't flush' is that the P-trap is likely clogged or obstructed, preventing proper drainage. This can happen due to buildup of hair, soap scum, or other debris in the trap. Symptoms may include slow draining, gurgling sounds, or water backing up in the sink or tub.
To fix a clogged P-trap, you can try using a plunger or a small plumber's snake/auger to dislodge the blockage. If that doesn't work, you may need to disassemble the P-trap and clean it out manually. However, if the clog is severe or you're not comfortable doing it yourself, it's best to call a professional plumber to inspect and clear the blockage.