There's a name for it in the Royal Academy (4)
I believe the answer is:
rita
'there's a name' is the definition.
'it in the royal academy' is the wordplay.
'in' means one lot of letters goes inside another.
'the royal academy' becomes 'ra'.
'it' inserted inside 'ra' is 'RITA'.
'for' is the link.
(Other definitions for rita that I've seen before include "-- Hayworth, US film actress" , "The Beatles' lovely girl" , "Educating - - , film" , "subject of education" , "Lovely -- , meter maid (Beatles)" .)