Pair off with student for a month (5)
I believe the answer is:
april
'a month' is the definition.
(April is an example)
'pair off with student' is the wordplay.
'off with' indicates anagramming the letters.
'student' becomes 'l' (as in L-plates for learner drivers).
'pair' mixed up with 'l' can make 'APRIL'.
'for' is the link.
(Other definitions for april that I've seen before include "''All in an ... evening / ... airs were abroad'' (by Katherine Tynan)" , "The month with All Fools' Day" , "Part of spring" , "''... is the cruellest month''(Elliot)" , "Start of the second quarter" .)