Yeah, it stays on for a minute or two before clicking off.
Suggest de-powering your pump, then fully drain your large galvanized hydro-pneumatic (H-P) pressure tank. Once fully drained, reactivate the pump to cause water to reenter and compress the appropriate volume of air into a smaller amount of space within the tank
A well supply pipe to an H-P pressure tank, will be usually equipped with a bleeder orifice to allow some air to be automatically added each time the pump becomes activated. Adding air on an ongoing basis is needed as the air within the tank is in direct contact with the water, which will cause some of the air to become dissolved into the water, thereby resulting in insufficient air remaining in the tank within a short period of time. When this occurs, the tank will become filled with too much water, which will result in the pump 'short cycling' as you describe.
In addition, the H-P tank will be commonly also equipped with an air volume control valve, which will cause excess air to be vented out so as to maintain the correct ratio of air to water within the tank. Short cycling' can also occur If there is too much compressed air within the tank, as that will limit the amount of water that will enter the tank.
As you said water flow ceases before the pump becomes activated, and the pump only operates each time for a short period, I suspect there is insufficient air in your pressure tank, so perhaps your issue is actually caused by the bleeder valve not adding sufficient air into the tank, or maybe, because you are supplying from a cistern, perhaps no bleeder valve is installed? Without sufficient air in the H-P tank, there will be too little compressed air stored to push all of the water out from the tank, resulting in rapid pressure drop and a flow dropout until the pump becomes activated and restores flow to fixtures.
Most modern well systems now utilize a captive air pressure tank as opposed to an H-P tank. A captive tank utilizes a flexible, sealed diaphragm to separate the air and water chambers, which then results in no air becoming dissolved in the water, and therefore, no bleeder valve or air volume control needed. In addition, as the sealed air chamber will be pre-charged with compressed air before any water enters the water chamber, this will allow a captive air pressure tank to be considerably smaller in physical size compared to an H-P tank with the same draw down (water) capacity.