Can i cast a spell after collecting rewards for an adventure?
The rules as written do not clarify one way or the other - it is up to a house ruling.  However, many cards have specific timing for them listed (such as "At the start of your turn" or "At the end of your turn").  With this in mind, for any card that does not have a timing listed, you could make a case for justifying its usage at any time during your turn, including immediately before it ends.