Can a cleanup happen while I’m finalizing a card or ability?

No. A cleanup cannot occur during the process of finalizing a card or ability.

Finalizing is part of the FEPR process—Finalize, Execute, Pass, Resolve—which is the process that handles items on the chain. Finalizing is the step where you make choices, pay costs, and check legality. Cleanups will occur after adding an item to the chain, after finalizing that item, and after that item resolves. At each point that a cleanup occurs, it will pause the FEPR process until it is complete. However, cleanups will only happen between these steps and never during a step itself.