It is possible that your "city water" may have contained grit "particles" [as Richard said], which may have caused both problems.
You should always
use an inline RV water filter in the hose, both
when filling the water tank, and when using city water directly, to keep grit out of the plumbing system. [link below]
For the pump, turn off the city water, make sure you have good clean water in your water tank, open a cold faucet, and then turn the pump on/off 4-5 times, about 4-5 seconds apart each time. This should free up any particles in the pump's anti-back-flow check valve. Close the faucet, turn on the city water supply, and see if the problem is solved. Repeat if needed.
For the black tank flush system, if grit got to the spray heads mounted in the top of the black water tank, and the tiny orifices in the spray heads are clogged with grit, those spray heads may have to be replaced, from what I have read here. This is a huge project IMO.
If you did get grit in the water somehow, there is probably some in the bottom of your water heater also. I would turn it off, let the water cool down so it is not scalding hot, remove the drain plug, open the PRV [pressure relief valve] and a hot faucet, and let the water heater tank drain. Have a bucket ready before you do this, and try to catch some of the initial water coming out of the drain hole -- is there visible grit in it?
Maybe you could review where your "city water" has come from recently. If it is from a private water system at a campground, which uses its own well water [vs. municipal water], that could be the source of the problem IMO. Even municipal city water can contain grit BTW, like then they work on the pipes in the street. In our neighborhood at home, we all monitor city water quality, and alert each other when anyone sees murky city water. One foul load of this can ruin a clothes washer full of nice white sheets etc..
Same solution for the trailer -- always use an inline water filter in the hose like this one: