I think that you have a leaky fitting on the suction side of the pump, between it and the water tank. When you run the water for a minute or so it sucks a bit of air in which builds up near the suction of the pump until the pump can't pump any more. It then clears the air through the pump. That is the sputtering you hear. Then it is ok for another minute or so.
Find and fix the leak and your problems should go away.
David
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