When I flush the toilet, tank doesn't refill. The water seems to leak out under the cap instead of going into the refill tube. But if I turn the water off, then back on, the surge of water seems to fix the problem, and water goes right to the refill tube and the tank fills back up. Any ideas on why it does this?

Replace the flush valve. Inexpensive easy DIY fix.

What ball cock you have? Do you wish to repair or replace? Back to you, Tom