Argue once mistakenly and inspire (9)
I believe the answer is:
encourage
'inspire' is the definition.
(I know that encourage is a type of inspire)
'argue once mistakenly' is the wordplay.
'mistakenly' is an anagram indicator.
'argue'+'once'='argueonce'
'argueonce' is an anagram of 'ENCOURAGE'.
'and' acts as a link.
(Other definitions for encourage that I've seen before include "Spur on, inspire" , "Give support, confidence or hope to" , "Give support to, stimulate hope" , "Spurs on" , "Give hope or confidence to, help to advance" .)