Champion, with something short of a bullseye (6)
I believe the answer is:
winner
'champion' is the definition.
(I know that champion can be written as winner)
'with something short of a bullseye' is the wordplay.
'with' becomes 'w' (abbreviation).
'something short of a bullseye' becomes 'inner' (area of a target close to the bullseye).
'w'+'inner'='WINNER'
(Other definitions for winner that I've seen before include "One that came first" , "film director" , "Champion" , "success" , "Top player" .)