Stubborn former student, say, about unfinished inn (9)
I believe the answer is:
obstinate
'stubborn' is the definition.
(synonyms)
'former student say about unfinished inn' is the wordplay.
'former student' becomes 'OB' (abbreviation for 'Old Boy').
'say' becomes 'state' (synonyms).
'about' is an insertion indicator.
'unfinished' means to remove the last letter.
'inn' with its final letter taken off is 'in'.
'state' going around 'in' is 'stinate'.
'ob'+'stinate'='OBSTINATE'
(Other definitions for obstinate that I've seen before include "Female bird" , "Stubborn, obdurate" , "Stubborn or unyielding" , "Mulish" , "such as a mule" .)