Stick around at home for the dog (6)
I believe the answer is:
canine
'the dog' is the definition.
(thesaurus)
'stick around at home' is the wordplay.
'stick' becomes 'cane' (cane is a kind of stick).
'around' is an insertion indicator.
'at home' becomes 'in' ('in' can be similar in meaning to 'at home').
'cane' enclosing 'in' is 'CANINE'.
'for' is the link.
(Other definitions for canine that I've seen before include "Of or like dogs" , "Fang" , "Dog-like" , "like a setter" , "Maybe setter" .)