Rewrite an overdue essay (9)
I believe the answer is:
endeavour
'essay' is the definition.
(I know that endeavour is a more specific form of the action essay)
'rewrite an overdue' is the wordplay.
'rewrite' indicates an anagram.
'an'+'overdue'='anoverdue'
'anoverdue' with letters rearranged gives 'ENDEAVOUR'.
(Other definitions for endeavour that I've seen before include "Determined attempt" , "Cook's old vessel" , "Attempt by employing effort" , "Try hard; enterprise" , "Morse" .)