|> the sink would pump ... (dirty) water.
I think that's probably your clue - dirt in the water system. Aboard our boat we found there were gribbledies (a technical term) floating in the water that the suction of the pump was pulling into the line. Once you stop pumping, they float away, hence the intermittent problem.
Start at the tank - open the inspection port and look inside for crud. If you can't look inside, try disconnecting the water line then filling the tank and letting it drain a couple of times - running a volume of water into the bilge may clear the tank. Keep doing this until the flow is free of crud. As to the pump line, I'd just hold a hose and the line (disconnected at both ends) together in my hand and let the hose pressure clear the line. Once you've done this (and mopped up all the water that's been sprayed all over in the process), try hooking the pump up to a source of clean water (a bucket thereof) and pumping until the pump is clear. In theory everything should now be okay, so hook the system up again and test. Good luck.
C&C 27 Mk IV