The air in a broken-down mine (4)
I believe the answer is:
mien
'the air' is the definition.
(thesaurus)
'a broken-down mine' is the wordplay.
'a broken down' indicates anagramming the letters.
'mine' with letters rearranged gives 'MIEN'.
'in' acts as a link.
(Other definitions for mien that I've seen before include "A face like mine" , "Counterance - mine?" , "Bearing, demeanour" , "The carriage" , "(One's) manner, bearing" .)