An extra note before end of course's recess (4)
I believe the answer is:
apse
'recess' is the definition.
(apse is a kind of recess)
'an extra note before end of course's' is the wordplay.
'an' becomes 'a'.
'extra note' becomes 'PS' (a PS or postscript in a letter).
'before' says to put letters next to each other.
'end of' says to take the final letters.
The final letter of 'course' is 'e'.
'a'+'ps'+'e'='APSE'
(Other definitions for apse that I've seen before include "Recess at east end of church" , "Rounded end of building" , "Semicircular recess in church" , "Where the altar is usually found in a church" , "Large recess in church" .)