To finish up best (3)
I believe the answer is:
top
'best' is the definition.
(top can mean the very best)
'to finish up' is the wordplay.
'finish' suggests halving the letters.
'up' halved is 'p'.
'to'+'p'='TOP'
(Other definitions for top that I've seen before include "Children's toy" , "Upper side" , "Toy; uppermost point" , "The circus tent can be a big one" , "jumper, say?" .)