Suddenly close being seen in public (9)
I believe the answer is:
overnight
'suddenly' is the definition.
(synonyms)
'close being seen in public' is the wordplay.
'close' becomes 'nigh' (similar in meaning).
'being seen in' means one lot of letters goes inside another.
'public' becomes 'overt' (both can mean well-known or unconcealed).
'nigh' inserted inside 'overt' is 'OVERNIGHT'.
(Other definitions for overnight that I've seen before include "Very quickly" , "Lightning" , "Until morning; suddenly" , "until tomorrow" , "Lasting from dusk to dawn" .)