Disgraceful mark idiots picked up on degree course (6)
I believe the answer is:
stigma
'disgraceful mark' is the definition.
(I've seen this before)
'on degree course' is the wordplay.
'on' means one lot of letters go next to another.
'degree' becomes 'MA' (Master of Arts is a type of degree).
'course' becomes 'stig' (I am not sure about this - if you are sure you should believe this answer much more).
'ma' put after 'stig' is 'STIGMA'.
'idiots picked up' is the link.
This may not be right. Some or all of it may be part of another bit of the clue.
Can you help me to learn more?
(Other definitions for stigma that I've seen before include "Disgraceful mark" , "evidence of nastiness?" , "A stain as on character" , "Blot" , "Brand or mark of infamy" .)