What pressure range does the pressure gauge show when the pump runs from on to off?
If that filter is between the pump and the pressure switch/tank, remove it before you blow the pump off the line in the well or the plumbing.
Have you checked the air pressure in the pressure tank, with no water in it, to be 1-2 psi less than the turn the pump on pressure switch setting? I.E 30 on 50 off on the gauge says the air pressure in the tank wit hno water in it should be 29-28 psi. That air pressure provides the power to move water when your pump is not running. The air pressure and your switch settings are adjustable. The higher pressure the more water you get in gpm. Your tank size is unimportant. Pumps are sized by the gpm the house requires and then the hp to get the job done. The 1/2 hp is probability just right for your well.
You need current water tests done for hardness, iron and pH at least.
Carbon/charcoal is not a good choice to remove dirt.