Coach old lecturer to sing (5)
I believe the answer is:
carol
'sing' is the definition.
(caroling is a kind of singing)
'coach old lecturer' is the wordplay.
'coach' becomes 'car' (as in a train coach or car).
'old' becomes 'o' (common abbreviation eg in OE for Old English).
'lecturer' becomes 'l' (abbreviation for lecturer).
'car'+'o'+'l'='CAROL'
'to' acts as a link.
(Other definitions for carol that I've seen before include "A traditional Christmas song or hymn" , "Female" , "Song celebrating the birth of Jesus" , "Religious song - girl's name" , "and 27: Waits" .)