Teacher with first of essays completed (4)
I believe the answer is:
done
'completed' is the definition.
(something that's been completed has been done)
'teacher with first of essays' is the wordplay.
'teacher' becomes 'don' (don can mean an academic - especially at Oxbridge).
'with' means one lot of letters go next to another.
'first of essays' becomes 'e' (dst letter of 'essays').
'don'+'e'='DONE'
(Other definitions for done that I've seen before include "Cooked; carried out" , "This sort of thing is socially acceptable" , "Finished or ended" , "Artist's workroom" , "Complete; finished" .)