There are a few things a home owner can do to lightly service a water well system.
Re-charging the air charge into the holding pressure tank or air bladder inside the tank may improve the situation, but unlikely. To increase the air charge in the pressure tank you need to maintain or add about 35 psi pressure into the tank or air bladder. A newer tank will have a rubber air bladder inside the tank like a blow-up ball that maintains pressure to the water flow even with varying pump operation. This bladder will require an air compressor to introduce air into it through a little valve that is similar to airing up a tire on a car.
This valve would be found near the pressure adjusting box on the immediate out-flow pipe from the tank.
Another more serious repair is more likely to explain the loss and return of water pressure. This solution will require a water well service company to remove your well pump to access this repair. The problem may be a small air leak on the suction side of your well pump and a leaky foot valve causing your pump to lose the water prime. This is at the well pump at the bottom of your well. This will require pulling the pump from the bottom of your well and is done by a service company. This problem with the air leak may cause a loss of water flow to your pressure tank from the well pump for longer than the 5 seconds. However the pressure tank continues flow during this loss until the pressure in the tank drops.