Regarding outspoken Irish writer's triumph (7)
I believe the answer is:
rejoice
'triumph' is the definition.
(synonyms)
'regarding outspoken irish writer's' is the wordplay.
'regarding' becomes 're'.
'outspoken irish writer's' becomes 'joice' (I can't explain this - if you can you should believe this answer much more).
're'+'joice'='REJOICE'
(Other definitions for rejoice that I've seen before include "Be exultant and glad" , "Exult, show delight" , "Be happy" , "Celebrate" , "Express delight" .)