The answer to 'P-trap won't flush' is that a clogged or blocked P-trap is the most common cause. The P-trap is the curved pipe under your sink that holds a small amount of water to prevent sewer gases from coming back up into your home. If this trap becomes clogged with hair, food, or other debris, it can prevent water from flowing through properly and cause a backup.
Some common signs of a clogged P-trap include slow draining, gurgling sounds, or a foul odor coming from the sink. To fix this, you'll first want to try plunging or using a snake/auger to clear the clog. 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 uncomfortable working on the plumbing, it's best to call in a professional plumber to diagnose and resolve the issue.