Hunger of a pet in mine close to one (8)
I believe the answer is:
appetite
'hunger' is the definition.
(thesaurus)
'a pet in mine close to one' is the wordplay.
'in' means one lot of letters goes inside another.
'mine' becomes 'pit' (down the pit).
'close to' says to take the final letters (the close/ending of).
The last letter of 'one' is 'e'.
'pet' put within 'pit' is 'ppetit'.
'a'+'ppetit'+'e'='APPETITE'
'of' is the link.
(Other definitions for appetite that I've seen before include "Hunger; desire" , "Natural desire to satisfy a need" , "stomach" , "Desire or liking, for power or food" , "Natural craving" .)