I think you need to be a bit more "specific" with your problem. Are you saying that you get "continuous spitting" when you leave the tap open "forever" (ie. a long time) in that, you never get a continuous steady flow. OR are you a saying that any time you open a tap, you get splitting but eventually get a continuous flow.
IF you never get "steady flow", like cause is a blocked intake or pump problem. IF you always get "spitting" during the initial open, then like a pump problem possibly related to a failure of the oneway valve / seal at the pump.
IS your pump always switched on? IF so, does it run when no one is using any water? If its isn't always switched on, when you do switch it on, does it initially run?