Let back in to study somewhere in Cambridge (7)
I believe the answer is:
readmit
'let back in' is the definition.
(I've seen this before)
'study somewhere in cambridge' is the wordplay.
'study' becomes 'read' (synonyms).
'somewhere in cambridge' becomes 'MIT' (MIT in Cambridge, Massachusetts).
'read'+'mit'='READMIT'
'to' acts as a link.
(Other definitions for readmit that I've seen before include "Dear Tim (anag.)" , "Again accept" , "Let back in" , "Let in a second time" , "Take in again" .)