Private hostelry has support of British monarch (5)
I believe the answer is:
inner
'private' is the definition.
(as in one's private or inner thoughts)
'hostelry has support of british monarch' is the wordplay.
I cannot really understand how this works, but
'hostelry' could be 'inn' (synonyms) and 'inn' is present in the answer.
'monarch' could be 'er' (abbreviation for Elizabeth Regina, British Queen) and 'er' is found in the answer.
This accounts for all the letters.
This explanation may well be incorrect...
Can you help me to learn more?
(Other definitions for inner that I've seen before include "Target next to bull" , "It's next to the bull on the board" , "Area near to bull's-eye" , "Good shot at archery" , "Concealed" .)