Stop by a Harbor Freight and by the $6.00 small red Multimeter.
As with most things at HF--it borders on a piece of junk--but it is useful.
With the Multimeter you can use Ohms (which the is Omega symbol) to measure resistance (1 end of a wire to the other end) and you can measure voltage.
As you probably know, your water pump runs on 12 volts, which come from the battery, you do NOT need to be plugged in work on your problem.
The tips on leads for the multimeter are very sharp, so as long as you can see the wires going to the pump, you can puncture the insulation of both wires at the same time to measure 12v. Set the dial on 20 DC Volts. Turn it on with the little slide switch.
You might be able to stick 1 or both probes into the connection on the pump, or if there is a crimp terminal on the wires to connect to the pump, you could disconnect them, you may need a pair of long nosed pliers to pull the terminals out of the pump.
Once you turn on the meter, the display will show 0. What you want to see is 12V, when you have voltage at the pump. Try it our across your battery in you car, truck trails just to see how it works.
Hopefully, as others have mentioned, If the pressure of a city water connection is higher than the pumps cut-off switch, the pump will not run. If no external water, the pump will start running when the water pressure falls below a preset level, probably around 35psi, and stop running when the pressure gets to 45psi.
If the trailer has been in a clod winter without being properly winterized--it is probable the pump is bad and will need replacing. Amazon sell lots of models.
If you provide some pix of the pump and wiring--it would be very useful to help you.