Bird of prey pursuing bishop’s dog (6)
I believe the answer is:
beagle
'dog' is the definition.
(beagle is a kind of dog)
'bird of prey pursuing bishop's' is the wordplay.
'bird of prey' becomes 'eagle' (eagle is a kind of bird of prey).
'pursuing' means one lot of letters go next to another.
'bishop' becomes 'B' (abbreviation for bishop in chess).
'eagle' after 'b' is 'BEAGLE'.
(Other definitions for beagle that I've seen before include "5 [MANS BEST FRIEND]" , "Short-haired dog" , "Darwin voyaged in this ship" , "Hunting hound" , "Small hunting dog" .)