Biden regularly on time visiting Conservative after no bad publicity (9)
I believe the answer is:
notoriety
'bad publicity' is the definition.
(notoriety is fame for a bad reason)
I don't understand how the remainder of the clue works.
(Other definitions for notoriety that I've seen before include "Bad publicity" , "Fame from bad deeds" , "Public exposure" , "Bad reputation" , "Infamy, disrepute" .)