Supervisor for hunting a bird in green space (4,6)
I believe the answer is:
game warden
'supervisor for hunting' is the definition.
(game are animals to be hunted)
'a bird in green space' is the wordplay.
'bird' becomes 'mew' (mew is a kind of bird).
'in' is an insertion indicator.
'green space' becomes 'garden'.
'a'+'mew'='amew'
'amew' inserted into 'garden' is 'GAME WARDEN'.
(Other definitions for game warden that I've seen before include "Wild animal guardian" , "Reserve keeper" , "reserve worker" , "Ranger" , "One employed to protect rhinos, say" .)