Person writing second message shortened by 60% (4)
I believe the answer is:
memo
'message shortened by 60' is the definition.
(short for memorandum - 60% of the term is removed)
'person writing second' is the wordplay.
'person writing' becomes 'me' (the person writing the clues).
'second' becomes 'mo' (both are short periods of time).
'me'+'mo'='MEMO'
(Other definitions for memo that I've seen before include "Reminder (abbr)" , "It will remind" , "- Currie, Br. politician" , "Brief note, usually between co-workers" , "Note written as a reminder" .)