Can a player cast a spell during the cleanup step?

Under normal circumstances, no player receives priority during the cleanup step, which means no one can cast spells or activate abilities during it. The cleanup step is a purely mechanical phase where the active player discards down to their maximum hand size and damage is removed from permanents (CR 514.1, 514.2).

Priority is only given to players when a state-based action is performed or a triggered ability triggers during the cleanup step (CR 514.3). If either of those things happens, players do receive priority and may cast spells or activate abilities, after which another cleanup step is performed to handle any new damage or hand-size issues.

So in the vast majority of games, the cleanup step simply happens automatically with no player interaction at all. It is one of the only steps in the game where priority is not routinely passed.

Concrete example: Your opponent ends their turn with 8 cards and discards down to 7 during cleanup. A trigger fires — say, a 'whenever a card is put into a graveyard' effect. Both players now receive priority and you could cast an instant. After the stack resolves, a new cleanup step begins.