The problem of rapid drop off in pressure that you discribe is almost always due to a waterlogged tank. Newer tanks have rubber bladders that allow the tank to fill with water and compress the air filled badder above the water to maintain a proper “Head of air” above the water in the tank. When these fail, it is usually due to a leak that lets water get into the bladder. the only fix in that case it to replace the defective tank.
Older water tanks depend on a “head” of air above the water that is compressed as the pump fills the tank with water. These tanks are inherently leaky (to the head of air) and need to be regularly checked. At full pressure, most of these tanks thould be about 2/3 full of water. You may be able to determine the water level by condensation on the outside of the tank or by tapping on the side of the tank. If such a tank gets waterlogged, you need to turn of the pump, drain the tank, and open the top of the tank to the atmosphere (you can do this by unscrewing the plug at the top of the tank…usually a pressure guage is fitted to the plug opening, in which case you unscrew the guage. Then reseal the plug (or screw the guage back in) and turn on the pump. That should fix a waterlogged tank. You should also check for air leaks in the presurized tank with a soapy solution (Rigid makes a good one, but a thick detergent/water mix will do). Fix any leaks. And if you have a non-bladder tank, get one.
Of course you should check that your pump cut-in/cut-out settings are correct, but that is rarely the problem.