Fail to notice writer's letter (7)
I believe the answer is:
missive
'letter' is the definition.
(synonyms)
'fail to notice writer's' is the wordplay.
'fail to notice' becomes 'miss' (I've seen this in other clues).
'writer's' becomes 'i've' ('I have' can refer to the person writing).
'miss'+'ive'='MISSIVE'
(Other definitions for missive that I've seen before include "message" , "intelligence in this?" , "Letter" .)