Let in again to study at American institute (7)
I believe the answer is:
readmit
'let in again' is the definition.
('admit' can be a synonym for 'let in' so perhaps 'readmit' is OK for 'let in again')
'study at american institute' is the wordplay.
'study' becomes 'read' ('read' can be a synonym of 'study').
'at' means one lot of letters go next to another.
'american institute' becomes 'mit' (I've seen this in other clues).
'read'+'mit'='READMIT'
'to' is the link.
(Other definitions for readmit that I've seen before include "Welcome back" , "Let back in" , "Take in again" , "Allow back into" , "Dear Tim (anag.)" .)