Songbird cut half of this fly (6)
I believe the answer is:
thrush
'songbird' is the definition.
(type of songbird)
'cut half of this fly' is the wordplay.
'cut half of' suggests halving the letters.
'fly' becomes 'rush' (fly can mean to rush around).
'this' with half the letters taken is 'th'.
'th'+'rush'='THRUSH'
(Other definitions for thrush that I've seen before include "Brown songbird" , "Songbird with brownish back and speckled breast" , "Common British songbird" , "Speckle-breasted songbird" , "Throstle" .)