Close to sleep, come to in lecture (6)
I believe the answer is:
preach
'lecture' is the definition.
(I know that lecture is the is a more specific form of the action of preach.)
'close to sleep come to' is the wordplay.
'close to' indicates one should take the final letters (the close of something is the end).
'come to' becomes 'reach' (I've seen this in another clue).
The final letter of 'sleep' is 'p'.
'p'+'reach'='PREACH'
'in' acts as a link.
(Other definitions for preach that I've seen before include "Give, eg, a sermon" , "Proclaim (religion)" , "Publicly proclaim" , "Lecture" , "Proclaim (a religious message)" .)