A US Institute’s upset about finally gaining mark of disgrace (6)
I believe the answer is:
stigma
'mark of disgrace' is the definition.
(I've seen this before)
'a us institute's upset about finally' is the wordplay.
'us institute's' becomes 'mits' (Massachusetts Institute of Technology).
'upset' is a reversal indicator.
'about' is an insertion indicator.
'finally' becomes 'g' (I can't justify this - if you can you should believe this answer much more).
'a'+'mits'='amits'
'amits' back-to-front is 'stima'.
'stima' going around 'g' is 'STIGMA'.
'gaining' acts as a link.
This may not be correct. It may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for stigma that I've seen before include "Brand or mark of infamy" , "evidence of nastiness?" , "A stain as on character" , "Mark of dishonour" , "Badge of shame" .)