A note that's added on the end of late recess (4)
I believe the answer is:
apse
'recess' is the definition.
(apse is a kind of recess)
'a note that's added on the end of late' is the wordplay.
'note' becomes 'ps' (a PS or postscript in a letter).
'that's added' says to put letters next to each other.
'on' means one lot of letters go next to another.
'the end of' suggests the final letters.
The final letter of 'late' is 'e'.
'a'+'ps'+'e'='APSE'
(Other definitions for apse that I've seen before include "inside church?" , "Large recess in church" , "In sacred space" , "Domed section of a church" , "Offshoot from the choir, nave or aisle in a church" .)