Promotion and Cambridge university grant (5)
I believe the answer is:
admit
'grant' is the definition.
(grant can mean to agree or admit something)
'promotion and cambridge university' is the wordplay.
'promotion' becomes 'ad' (short for advert).
'and' says to put letters next to each other.
'cambridge university' becomes 'MIT' (MIT in Cambridge, Massachusetts).
'ad'+'mit'='ADMIT'
(Other definitions for admit that I've seen before include "Let in - concede" , "Take on" , "Agree to the truth" , "Come clean" , "Recognise" .)