The writer’s keeping a note cordial (5)
I believe the answer is:
matey
'cordial' is the definition.
'the writer's keeping a note' is the wordplay.
'the writer's' becomes 'my' (belonging to the person who set the crossword).
'keeping' is an insertion indicator.
'note' becomes 'te' (**).
'a'+'te'='ate'
'my' placed around 'ate' is 'MATEY'.
(Other definitions for matey that I've seen before include "uncomfortably friendly" , "Chummy" , "china" , "Close chum (slang)" , "Pirate's chum (colloq.)" .)