Woman runs business, one conveying messages by air (7,6)
I believe the answer is:
carrier pigeon
'one conveying messages by air' is the definition.
'woman runs business' is the wordplay.
'woman' becomes 'carrie' ().
'runs' becomes 'r' (cricket abbreviation).
'business' becomes 'pigeon' (pigeon can mean someone's business or affairs).
'carrie'+'r'+'pigeon'='CARRIER-PIGEON'
(Other definitions for carrier pigeon that I've seen before include "Messenger bird" , "Aerial messenger" , "Bird used for communication" , "Avian messenger" , "Bird with message" .)