A bird. Mine, thanks! (5)
I believe the answer is:
pitta
'a bird' is the definition.
(pitta is a kind of bird)
'mine thanks' is the wordplay.
'mine' becomes 'pit' (down the pit).
'thanks' becomes 'ta' (UK informal version of 'thank you').
'pit'+'ta'='PITTA'
(Other definitions for pitta that I've seen before include "Slightly leavened bread" , "Type of flat bread that's split to hold a filling" , "Partially-leavened bread of the Near East" , "Greek unleavened bread" , "(E.g. Greek) bread" .)