water pumps and irrigation systems?

So im trying to get my irrigation up and runing I have a decent size yard in the southern united states. Problem is i have no steady supply of water for it so planting is near impossible. If im not irrigating  it turns to sand.

Long story short I have a old decrepid water pump that im preatty certain is not working any more. At first it was thought that the pump was not geting charged because there was no back flow valve between the pump and water source so a back flow valve latter.

Im noticing as im hose charging the pump one of the heads coming on but the others hardly giving a dribble. between this head and the other heads is what i think is a zone splitter. At first I thought that the head between the zone splitter and the pump was stealing the charge as it was operating at full pressure. But once i cut the hose off to stop the manual charge of the pump the head was no longer operating. So my line of thinking is that if the pump is runing and is not pulling from the water source then the pumps not working.Is this the correct line of thinking I know jack and all about water pumps and irrigation systems so any help would be greate.