Bishop adopting star competitor (5)
I believe the answer is:
racer
'competitor' is the definition.
(thesaurus)
'bishop adopting star' is the wordplay.
'bishop' becomes 'rr' (abbreviation for Right Reverend).
'adopting' is an insertion indicator.
'star' becomes 'ace' (synonyms).
'rr' going around 'ace' is 'RACER'.
(Other definitions for racer that I've seen before include "Runner" , "someone driving very fast?" , "One car" , "Fast driver" , "American snake" .)