It sounds to me like you have a holding tank that has a bladder in it to separate the air from the water and hold the air precharge.
From what you described, the air precharge is set to 30 psi and the "on" pressure for the pump is set for 20 psi. This is not good, as the tank will run out of water before the pump kicks back on. Thus the water pressure drops way off just as the pump restarts.
You need to shut off the power to the pump, open up a faucet until water completely stops running out. Then check the precharge of air in the tank. It should be approximately 10 psi for your pressure switch which is set to turn on at 20 psi. Bleed air out or add air to get the 10 psi. Leave the faucet open while you do this.
After the precharge is set for 10 psi, turn the power back on.