Off-the-cuff claim to be on time at university (9)
I believe the answer is:
impromptu
'off-the-cuff' is the definition.
(I know that off-the-cuff can be written as impromptu)
'claim to be on time at university' is the wordplay.
'claim to be' becomes 'I'm' (eg 'I'm fine' is a claim to be fine).
'on time' becomes 'prompt' (synonyms).
'at' means one lot of letters go next to another.
'university' becomes 'u' (abbreviation for university).
'im'+'prompt'+'u'='IMPROMPTU'
(Other definitions for impromptu that I've seen before include "Spontaneous" , "Ad-lib" , "Unplanned; unrehearsed" , "Offhand" , "On the spur of the moment" .)