You can remove a stuck tap cartridge with a special cartridge-pulling tool available at home improvement centres and some plumbing supply houses. The process will be slow but you want to remove the stuck cartridge without damaging any of the copper plumbing.

Turn the thumbscrew on the top middle of the cartridge-pulling tool counter-clockwise with your fingers. Place the cartridge-pulling tool over the front portion of the tap cartridge. Twist the cartridge-pulling tool clockwise until the tool locks over the cartridge.

Turn the thumbscrew on the top middle of the cartridge-pulling tool counter-clockwise with your fingers.

Place the cartridge-pulling tool over the front portion of the faucet cartridge.

Turn the thumbscrew clockwise until you cannot turn the screw. Tighten the nut on the centre of the cartridge-pulling tool. Pull the tool and the cartridge straight out of the tap plumbing. Pour some vinegar into the tap to loosen and dissolve any calcium deposits.

Once the cartridge can be turned, grab the end of your cartridge with a pair of vice grips and pull the stuck cartridge out of the tap plumbing.