Auntie's book on queen, perhaps (4)
I believe the answer is:
beeb
'auntie's' is the definition.
(both are nicknames for the BBC)
'book on queen perhaps' is the wordplay.
'book' becomes 'b' (abbreviation for book).
'on' means one lot of letters go next to another.
'queen perhaps' becomes 'bee' (queen bee is a type of bee).
'b' put after 'bee' is 'BEEB'.
(Other definitions for beeb that I've seen before include "Old microcomputer" , "BBC, familiarly" , "Informally, the British broadcasting service" , "BBC (informally)" , "BBC (colloq" .)