Like dog’s home, stick around! (6)
I believe the answer is:
canine
'like dog's' is the definition.
(canine can mean relating to dogs)
'home stick around' is the wordplay.
'home' becomes 'in' ('he's home' can mean 'he's in').
'stick' becomes 'cane' (cane is a kind of stick).
'around' is an insertion indicator.
'in' going into 'cane' is 'CANINE'.
(Other definitions for canine that I've seen before include "Boxer" , "Tooth; dog" , "Relating to dogs" , "A tooth; of our best friend" , "such as Lassie" .)