She sponsors France’s best exhibitions at first, somehow (12)
I believe the answer is:
benefactress
'she sponsors' is the definition.
'france's best exhibitions at first somehow' is the wordplay.
'at first' says to take the initial letters.
'somehow' indicates an anagram.
The initial letter of 'exhibitions' is 'e'.
'frances'+'best'+'e'='francesbeste'
'francesbeste' anagrammed gives 'BENEFACTRESS'.
(Other definitions for benefactress that I've seen before include "Generous female donor" , "Lady patron" , "She provides financial support" , "one who does good" , "Generous woman" .)