Bird not finishing off the fly (6)
I believe the answer is:
thrush
'bird' is the definition.
(thrush is a kind of bird)
'not finishing off the fly' is the wordplay.
'not finishing off' means to remove the last letter.
'fly' becomes 'rush' (fly can mean to rush around).
'the' with its final letter taken away is 'th'.
'th'+'rush'='THRUSH'
(Other definitions for thrush that I've seen before include "Infection of the mouth and throat" , "Brownish songbirds" , "Small bird - fungal mouth infection" , "Throstle" , "Common singing bird or throat infection" .)