If you don't want to replace the faucet nor drill out the screen, disconnect the water lines, open the both valves full and then blow it backwards with only medium to light pressure. Preferrably with a water hose which will wash more of the trash out. Make sure to catch the little bit of water and debris a towel or bucket.
Leave the valves open, reconnect the water lines and then turn the pump on. You want to run a little bit of water through the valves to clean any debris before closing them or you risk messing up the seal.