I don't know the exact term for the part I'm referring to so I'll describe it. The faucet's actual closing is done by something similar to this but without the slits for water. I assume that's called the cartridge.
It presses against a part inside the faucet that is basically the ending of a pipe.
The end of the cartridge is like a disk of rubber (with no hole), and the pipe is metal.
The problem is that the pipe somehow got a small groove (from the inside to the outside, though on its ending surface) so though the cartridge stops water from coming straight out the pipe, there is some coming out "sideways", which manifests itself externally, as a slow drip.
Pipe-ending and groove (exaggerated):
Closing the faucet handle very tight stops that drip, but ruins the rubber part, and after some time, I need to buy a new cartridge.
Is there a way of sealing the groove or the fixing the rubber disk, or do I have to buy a new faucet? This is for drinking/cooking, so I don't want to seal it with something that might come off.