Perhaps on air one refused a farewell (8)
I believe the answer is:
sayonara
'farewell' is the definition.
(I know that sayonara is a type of farewell)
'perhaps on air one refused a' is the wordplay.
'perhaps' becomes 'say' (both can mean 'for example').
'one' becomes 'i' (Roman numeral).
'refused' suggests deleting specific letters.
'air' with 'i' removed is 'ar'.
'say'+'on'+'ar'+'a'='SAYONARA'
(Other definitions for sayonara that I've seen before include "Japanese farewell" , "So long" , "Goodbye or farewell in Japan" , "Japanese goodbye" , "Japanese valediction" .)