Hunting dog beginning to bother bird of prey (6)
I believe the answer is:
beagle
'hunting dog' is the definition.
(I know that beagle is a type of hunting dog)
'beginning to bother bird of prey' is the wordplay.
'beginning to' suggests taking the first letters.
'bird of prey' becomes 'eagle' (eagle is a kind of bird of prey).
The initial letter of 'bother' is 'b'.
'b'+'eagle'='BEAGLE'
(Other definitions for beagle that I've seen before include "Small, sturdy breed of hound" , "hunting animal with smooth coat" , "Hunting dog followed on foot" , "Hound dog" , "Small short-legged hound" .)