Pursues pets (4)
I believe the answer is:
dogs
This clue is a double definition.
'pursues' is the first definition.
(dogging is a kind of pursuing)
'pets' is the second definition.
(dogs are often kept as pets)
(Other definitions for dogs that I've seen before include "pets" , "Canines" , "Persistently follows" , "eg 27 terriers" , "Hounds" .)