Old lecturer enters competition with authority (6)
I believe the answer is:
oracle
'authority' is the definition.
(oracle can mean an authority on a topic)
'old lecturer enters competition' is the wordplay.
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'lecturer' becomes 'l' (abbreviation for lecturer).
'enters' means one lot of letters goes inside another.
'competition' becomes 'race' (race is a kind of competition).
'l' going into 'race' is 'racle'.
'o'+'racle'='ORACLE'
'with' acts as a link.
(Other definitions for oracle that I've seen before include "Place where ancient Greeks consulted gods, as at Delphi" , "Shrine; its message" , "O, clear off for the Greek wise man" , "Agency of divine revelation" , "Person regarded as source of knowledge, like old Greek prophet" .)