Challenge mission no-one's backing (8)
I believe the answer is:
question
'challenge' is the definition.
(I've seen this before)
'mission no-one's backing' is the wordplay.
'mission' becomes 'quest' (I've seen this before).
'one' becomes 'i' (Roman numeral).
'backing' says the letters should be written in reverse.
'no'+'i'='noi'
'noi' backwards is 'ion'.
'quest'+'ion'='QUESTION'
(Other definitions for question that I've seen before include "Enquiry; doubt" , "Poser" , "Ask; doubt" , "Challenge" , "Query" .)