Turning on water pump while connected to city water won't hurt anything...won't really help anything either.
City water and pump discharge go to same main cold water line which then has various connections going to faucets/shower/toilet coldside and to water heater for hotside.
Which ever one has the higher line pressure is the one that will supply the plumbing system.
Typical water pump pressure is 45# (some have 55#) so if city water pressure is higher.....pump won't run even if turned on. If city water pressure is lower pump will run and city water will not flow due to a check valve at city water inlet connection that would close due to higher pump pressure.

Low flow at a faucet (if both hot & cold) is usually due to a clogged aerator screen.
Low flow at faucet on cold side could be due to water filter plugging up (in kitchen cabinet)