Finish off better stand-up routine (6)
I believe the answer is:
improv
'stand-up routine' is the definition.
'finish off better' is the wordplay.
'finish off' means to remove the last letter.
'better' becomes 'improve' ('improve' can be a synonym of 'better').
'improve' with its final letter taken off is 'IMPROV'.
(Other definitions for improv that I've seen before include "little ad-libbing" , "Off-the-cuff comedy" , "off-the-cuff humour" , "Stand-up comedy, perhaps" .)