Welcomed by Conservatives, personal service (7)
I believe the answer is:
vespers
'service' is the definition.
(vespers is a church service)
'welcomed by conservatives personal' is the wordplay.
'welcomed by' indicates a hidden word.
'VESPERS' can be found hidden inside 'conservatives personal'.
(Other definitions for vespers that I've seen before include "Evening service" , "Evening prayers" , "Evensong - RC service" , "An hour of worship" .)