Rapid cycling of a pump with a pressure tank is usually caused by waterlogging of the tank. There are two types of pressure tanks, bladder and non.
With a non-bladder tank, there's no separation of air and water. Water on the bottom, and air on top. The idea here is to have the water level almost at the bottom of the tank when the pump kicks on. Usually, there's a schrader (tire-type) of fitting somewhere on the tank, usually on top.
To precharge this type of tank, shut the pump off, and bleed all the water out of the tank. If you put air in while the discharge valve is open, it'll speed it up a bit. When the tank is empty (air coming out of the discharge), shut the discharge valve. Fill the tank with air until the pressure is a few PSI below where the pump would start. Start the pump, now you're good.
Bladder tanks have a rubber diaphragm in the center of the tank. Water and air never touch each other. The bladder expands and contracts with pressure change. It's possible that the bladder has ruptured. All bladder tanks have a schrader valve on top of the tank. Bleed some air from this valve. If any water comes out, the bladder is ruptured, and the tank will need to be replaced. To precharge a bladder tank, shut the pump off and bleed the water out. Add air (ok to do so during the bleeding) to about 2 PSI below where the pump would start. Start the pump. The pressure will jump right up to the precharge PSI, and fill from there. Unfortunately, if a bladder tank loses its precharge, it's usually because the bladder is ruptured, though sometimes the schrader valve leaks.