Man's mistake: arithmetic presumably dropped by learner (5)
I believe the answer is:
errol
'man's' is the definition.
'mistake arithmetic presumably dropped by learner' is the wordplay.
'mistake' becomes 'err' (to err is to make a mistake or blunder).
'arithmetic presumably' becomes 'R' (one of the three R's of education).
'dropped' is a deletion indicator.
'by' becomes 'ro' (I can't justify this - if you can you should believe this answer much more).
'learner' becomes 'l' (as in a learner driver with L-plates on their car).
'err' with 'r' removed is 'er'.
'er'+'ro'+'l'='ERROL'
(Other definitions for errol that I've seen before include "-- Flynn, Hollywood actor" , "Old film star" , "- Flynn, twentieth century film actor" , "- Flynn, US film actor" , "- - Flynn, film star" .)