Pickets court about rent supplement (10)
I believe the answer is:
postscript
'supplement' is the definition.
(I know that supplement can be written as postscript)
'pickets court about rent' is the wordplay.
'pickets' becomes 'posts' (picket can mean a post or stake).
'court' becomes 'ct' (abbreviation used in road names).
'about' means one lot of letters goes inside another.
'rent' becomes 'rip' (rent can mean a tear in something).
'ct' going around 'rip' is 'cript'.
'posts'+'cript'='POSTSCRIPT'
(Other definitions for postscript that I've seen before include "PS" , "Last added words" , "belated comment" , "Extra lines" , "part of letter" .)