Yes you can remove the filter cartridge and leave the filter running, however, you won't have any filter bacteria to remove ammonia, which will feed the algae bloom.
If you remove carbon from a filter, then throw the carbon away and do not replace it.
Filter bacteria living on a sponge or cartridge that is removed from the aquarium, will die when the sponge/ cartridge dries out (usually within a couple of hours).
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The easiest way to fix a green water problem is by doing 75-90% water changes each day until it clears, and reducing the light. Adding floating plants will help too.
Some people add live Daphnia or Rotifers to the tank and they eat the algae and help clear the water up. If you have fish in the tank, the Daphnia can be put into a breeding net where the green water can flow through the net but the Daphnia can't get out or be eaten by the fish.
Fish don't mind being in green water and it is fine for them as long as there is some aeration/ surface turbulence.