Politician vexed over new church stopping alcoholic drink (5-7)
I believe the answer is:
cross-bencher
'politician' is the definition.
(I know that cross-bencher is a type of legislator)
'vexed over new church stopping alcoholic drink' is the wordplay.
I cannot really see how this works, but
'vexed' could be 'cross' (both can mean angry) and 'cross' is located in the answer.
'new' could be 'n' (common abbreviation eg NT for New Testament) and 'n' is present in the answer.
'church' could be 'ch' (common abbreviation) and 'ch' is present in the answer.
'drink' could be 'beer' (beer is a kind of drink) and 'beer' is found in the remaining letters.
No letters remain.
This explanation may well be incorrect...
Can you help me to learn more?
(Other definitions for cross-bencher that I've seen before include "Independent member of Parliament" , "Independent House of Lords member" , "Independent or neutral MP" .)