Well-to-do island ruler meets under-secretary (10)
I believe the answer is:
prosperous
'well-to-do' is the definition.
(synonyms)
'island ruler meets under-secretary' is the wordplay.
'island ruler' becomes 'prospero' (I am not sure about this - if you are sure you should believe this answer much more).
'meets' is a charade indicator (letters next to each other).
'under secretary' becomes 'us'.
'prospero'+'us'='PROSPEROUS'
Can you help me to learn more?
(Other definitions for prosperous that I've seen before include "Well-to-do" , "Wealthy" , "Rich" , "Affluent" , "thriving" .)