Mark prosecuted about having continued (9)
I believe the answer is:
sustained
'continued' is the definition.
(sustaining is a kind of continuing)
'mark prosecuted about' is the wordplay.
'mark' becomes 'stain' (synonyms).
'prosecuted' becomes 'sued' (I've seen this in another clue).
'about' indicates putting letters inside.
'stain' put into 'sued' is 'SUSTAINED'.
'having' is the link.
(Other definitions for sustained that I've seen before include "Steady" , "Steadily maintained" , "Kept (something) going" , "held up" , "bore" .)