Become tedious letter-writer, by the sound of it? (4)
I believe the answer is:
pall
'become tedious' is the definition.
(to pall is to become dull)
'letter-writer by the sound of it?' is the wordplay.
'letter-writer' becomes 'paul' (Paul, biblical author of several epistles).
'by the sound of it?' indicates a 'sounds like' (homophone) clue.
'paul' sounds like 'PALL'.
(Other definitions for pall that I've seen before include "Cloth spread over a coffin" , "Cloy" , "Become tiresome" , "Become wearisome" , "Become boring and uninteresting" .)