Private hotel on lively Rome street (9)
I believe the answer is:
innermost
'private' is the definition.
(thesaurus)
'hotel on lively rome street' is the wordplay.
'hotel' becomes 'inn' (inn is a kind of hotel**).
'on' indicates putting letters inside.
'lively' indicates anagramming the letters (I've seen this in other clues).
'street' becomes 'st'.
'rome' is an anagram of 'ermo'.
'inn' placed into 'ermo' is 'innermo'.
'innermo'+'st'='INNERMOST'
(Other definitions for innermost that I've seen before include "Closest to the centre" , "Sort of secrets" , "Private" , "nearest the centre" , "Furthest in" .)