When can cards/power ups be played in King of Tokyo (and Power Up! expansion)?
The King of Tokyo rules only define two types of cards, Keep and Discard. (Page 2). There is no such thing as Temporary or Permanent cards. Many cards seem to represent power ups of the Monsters, but this isn't a game term.

Card Description
How to play the card is indicated above the description of the effect.
KEEP: Monsters keep these cards for the remainder of the game (sometimes the card will indicate it is discarded under certain circumstances.)
DISCARD: Monsters immediately resolve these then discard them.

As the Card Description indicates, the cards themselves will indicate when they can be played, with the exception of DISCARD cards that are played immediately after being purchased. Unless a card notes otherwise, you may play it and gain its effects immediately.
Permanent/Temporary Evolution cards can be played at any time, barring a timing restriction printed on the card. From the expansion rules (page 2)

Evolutions - Monsters keep their evolutions secret until they want to play them – which can be at any time. There are two kind of Evolutions:
Temporary Evolution: They are discarded after being played.
Permanent Evolution: They stay in play like a Keep card from the base game.