Only use the pump after you "sterilize" the water tank. There is a proceedure for this (baking soda, I think) that might be in your owners manual. If not, ask the dealer...or even ask here. You don't want contaminated water pumping through-out your system.
Absolutely no reason to turn on the pump when hooked up to "city" water. Most likely your problem is, as mentioned before a clogged strainer. My motorhome, there is no filter to the kitchen sink. However, there is one to the goose-neck on the other sink. When new, winterizing the motorhome I had to remove the filter and install the by-pass PVC pipe. (Winterizing would kill the filter.)I never replaced the filter. All's well with 13 years of travel.