Express regret in public for lapse (9)
I believe the answer is:
oversight
'for lapse' is the definition.
(synonyms)
'express regret in public' is the wordplay.
'express regret' becomes 'sigh' (a sigh expresses regret).
'in' is an insertion indicator.
'public' becomes 'overt' (both can mean well-known or unconcealed).
'sigh' inserted inside 'overt' is 'OVERSIGHT'.
(Other definitions for oversight that I've seen before include "act of negligence" , "Carelessness" , "what supervisor has" , "Failure to notice - blunder" , "Slip-up" .)