Private hotel on lively Rome street (9)
I believe the answer is:
Here is my best explanation:
'private' is the definition.
(I've seen this before)
'hotel on lively rome street' is the subsidiary indication.
'hotel' becomes 'inn' (inn is a kind of hotel).
'on' is a charade indicator (letters next to each other).
'lively' indicates an anagram (I've seen this in other clues).
'street' becomes 'st'.
'rome' anagrammed gives 'ermo'.
(Other definitions for innermost that I've seen before include "Private", "Closest to the centre" and "Furthest in".)