In Smash Up, do I have to execute an effect when a card is played or can I choose to do it later in the turn?
The effect resolves on the spot, giving you a floating extra play. However, if it's not your main play phase (i.e. specials during scoring) you must use the extra play immediately.

From the most recent rulebook (Pretty Pretty Smash Up):


  Extra: An extra minion or action is one you can play in addition to the normal quota of one of that type per turn. Extra cards are optional, and you don’t have to play the extra card right away, unless the extra card was gained by a Special ability or outside the Play Cards phase.


This is explicitly different during scoring:


  If you get to play extra minions or actions while a base is being scored, you must play those extra cards immediately or not at all.