Hermitage Museum's opening supporting a quiet artist (6)
I believe the answer is:
ashram
'hermitage' is the definition.
(I have seen 'Indian hermitage ' mean 'ashram' so perhaps 'hermitage' could also mean 'ashram')
'museum's opening supporting a quiet artist' is the wordplay.
'opening' indicates taking the first letters.
'supporting' is a charade indicator (letters next to each other) (in a down clue, some letters go below others).
'quiet' becomes 'sh' (**).
'artist' becomes 'ra' (Royal Academician, member of Royal Academy of Arts).
The initial letter of 'museums' is 'm'.
'a'+'sh'+'ra'='ashra'
'm' put after 'ashra' is 'ASHRAM'.
(Other definitions for ashram that I've seen before include "Place for religious retreat (in India)" , "Hindu place of retreat" , "Hindu place of religious learning located in a marsh" , "Place of religious retreat for Hindus" , "Indian hermitage" .)