Mark of disgrace that's initially concealed by Greek character (6)
I believe the answer is:
stigma
'mark of disgrace' is the definition.
(I've seen this in another clue)
'that's initially concealed by greek character' is the wordplay.
'initially' suggests taking the first letters.
'concealed by' indicates putting letters inside.
'greek character' becomes 'sigma' (letter from the Greek alphabet).
The first letter of 'thats' is 't'.
't' placed inside 'sigma' is 'STIGMA'.
(Other definitions for stigma that I've seen before include "Blot" , "Part of a flower" , "evidence of nastiness?" , "Mark of dishonour" , "Bad mark" .)