Make a mistake with assignment (6)
I believe the answer is:
errand
'assignment' is the definition.
(I have seen 'Short assignment ' mean 'errand' so perhaps 'assignment' could also mean 'errand')
'make a mistake with' is the wordplay.
'make a mistake' becomes 'err' ('err' can be a synonym of 'make a mistake').
'with' becomes 'and' (I've seen this before).
'err'+'and'='ERRAND'
(Other definitions for errand that I've seen before include "Short delivery journey" , "Messenger's job" , "Gone up" , "Job usually involving a short trip" , "run one" .)