Bishop has a bird and a dog (6)
I believe the answer is:
beagle
'a dog' is the definition.
(I know that beagle is a type of dog)
'bishop has a bird' is the wordplay.
'bishop' becomes 'B' (chess abbreviation).
'has' says to put letters next to each other.
'a bird' becomes 'eagle' (eagle is a type of bird).
'b'+'eagle'='BEAGLE'
'and' is the link.
(Other definitions for beagle that I've seen before include "HMS - was Darwin's ship; hunting dog" , "Short-haired dog" , "Small hunting dog" , "Hunting hound" , "Hound dog" .)