Air supplier in awfully hot minutes without oxygen (9)
I believe the answer is:
tunesmith
'air supplier' is the definition.
(someone who writes tunes or airs)
'awfully hot minutes without oxygen' is the wordplay.
'awfully' is an anagram indicator.
'without' indicates named letters should be taken away.
'oxygen' becomes 'O' (O is the chemical symbol for oxygen).
'hot'+'minutes'='hotminutes'
'hotminutes' with 'o' taken away is 'htminutes'.
'htminutes' anagrammed gives 'TUNESMITH'.
'in' is the link.
(Other definitions for tunesmith that I've seen before include "Composer" , "I produce strains" , "Melody Maker" .)