John Bull's back in wealthy English institution (5)
I believe the answer is:
elton
'john' is the definition.
(musician Elton John)
'bull's back in wealthy english institution' is the wordplay.
'back' says to take the final letters.
'in' indicates putting letters inside.
'wealthy english institution' becomes 'Eton' (English public school).
The final letter of 'bull' is 'l'.
'l' going into 'eton' is 'ELTON'.
(Other definitions for elton that I've seen before include "Ben --; -- John" , ". . . . . John (musician)" , "First name of singer/songwriter born Reginald Dwight in 1947" , "Let on he's John, the singer" , "Sir ___ John, singer-songwriter" .)