Make a mistake with assignment (6)
I believe the answer is:
Here is my best explanation:
'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).
(Other definitions for errand that I've seen before include "Short journey to get or bring", "Short assignment", "Delivery journey", "Short mission -- ran red (anag)", "Task involving trip", "Messenger's job", "A short journey to deliver orget goods", "run one", "Short journey to carry message for Renard" and "In purposeful trip".)