Growth of old academic institution, beginning again (5)
I believe the answer is:
polyp
'growth' is the definition.
(I know that polyp is a type of growth)
'old academic institution beginning again' is the wordplay.
'old academic institution' becomes 'poly' (short for polytechnic).
'beginning again' becomes 'p' (I can't explain this - if you can you should believe this answer much more).
'poly'+'p'='POLYP'
'of' acts as a link.
(Other definitions for polyp that I've seen before include "Small tumour" , "Sea creature; small growth" , "Coral form" , "Sea anemone, perhaps" , "Small growth (medical)" .)