Fed up by the end of Monday’s challenge (4)
I believe the answer is:
defy
'challenge' is the definition.
(defying is a kind of challenging)
'fed up by the end of monday's' is the wordplay.
'fed up' becomes 'def' ('fed' going up in a down clue).
'by' means one lot of letters go next to another.
'the end of' says to take the final letters.
The last letter of 'monday' is 'y'.
'def'+'y'='DEFY'
(Other definitions for defy that I've seen before include "Openly resist or refuse to obey" , "Openly resist (authority etc)" , "Refuse to obey (person)" , "Stand against" , "Disobey boldly and openly" .)