Time for an upgrade. :B
The pump is usually within a few feet of the fresh tank. These pumps can dry prime, but they aren't miracle workers.
SHURflo says place within six feet of water source for their RV pumps.
If I had the extra cash, I would pull it and replace with one similar or better. Just make sure both pumps use the same fittings, and it should be an easy swap.
Then you can find out what is wrong with this one, fix it and keep for a spare.
Once you pull it, you could wire it to a 12V battery, scrounge some tubing or hose and play with it until you figure out what is wrong.
SHURflo makes repair kits for most of their pumps. You have to figure out what is wrong before you know which repair kit to buy. It's hard to find small devices that have repair kits available these days, but the nature of the pump world leads most makers to provide them. Many of these pumps (or similar types) are used in the marine world. Smart skippers want to stock replacement parts and be able to fix stuff on the water.
As to your pump, my guess is the pressure switch or check valve. The design is such that when it senses a loss in pressure, it turns on and pumps. When it senses enough pressure (usually set at 65psi from factory), it turns off.
You could have a loose electrical connection, but if the delay between "on then off" is about the same every time, my guess is pressure related. If the vibration of the pump were cutting it out, I would guess the time lapse before turn off would be much more variable.
Good luck