Carbon does help to eliminate DOCs (dissolved organic carbon) and other impurities that the filter misses. In our low-bioload Betta setups, we counter these by a weekly 50% water change.
in my opinion, and that of many experienced keepers around here, carbon is expensive and irrelevant.
This should also be your last
cycling experience. From now on, when you start up a new tank, you'll have live nitrifying bacteria to kick-start your new cycle.