Champion missing with bull's-eye, almost (5)
I believe the answer is:
inner
'bull's-eye almost' is the definition.
(the circle outside the bull's-eye on a target)
'champion missing with' is the wordplay.
'champion' becomes 'winner' (winner of a competition).
'missing' indicates named letters should be taken away.
'with' becomes 'w' (abbreviation).
'winner' with 'w' taken out is 'INNER'.
(Other definitions for inner that I've seen before include "Area near to bull's-eye" , "may not satisfy this man" , "personal" , "Nearer the middle" , "Private, not expressed" .)