Behind the city water hose connection, behind that screen, is a check valve. It's a little button right in the very middle. Push that button with your finger. Remove that screen if it's there. See if that button moves freely. If it does not, then it's stuck.
Click here for a picture of what we're talking about.
The water pressure coming into the trailer has to be high enough to force that little spring loaded check valve to open and allow water to flow. If the incoming pressure is not high enough, water will not flow because there is not enough pressure to push that valve open.
About pressure tanks.
We are on our own well too. There is a pressure tank under our house (a big blue thing). It has a bladder inside it filled with air under pressure. On the opposite end water fills the tank until the pressure equalizes (to whatever the pressure is set.) When that happens, the water pump turns off.
When you turn a faucet on inside the house, the water will start flowing. The pressure tank will deplete the pressure, allowing the water to flow for some time before the water pump turns on again to build up pressure. If you do not have this pressure tank, the pump would be clicking on and off instantly as there is no room for pressure to build in pipes only.
Now, some people install smaller versions of these inside their RV. They can be as small as 2 gallons to 5 gallons, or as big as they can go to fit available space inside their camper. The purpose is exactly the same as in a house. It's to keep the water pump in the camper from cycling on and off so fast.
If you have a pressure tank in the camper AND in the house for the water source, then that could account for highs and lows in water pressure as you run water. But, if it's blocked and no water is running, then more than likely, it's that valve where the garden hose connects.
One thing you can do is put water in your fresh water tank. Turn off city water (disconnect the garden hose). Now turn on your water pump. Your water is now pumping from the fresh water tank and will not go through that check-valve. If the water flows steady with the water pump running, then you know the source of the problem is upstream of the water pump .... namely, the city water connection (garden hose connection).
That is the easiest way to determine if something inside the trailer is blocking any plumbing behind the flow of the water pump. Again, if water flows freely with the water pump, then the problem has to be that garden hose connection valve.