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50A - Unorthodox religions embraced by established church adviser (11). C?N?I?L?E?E

33A - Bronze pater with child bust (5, 5). T?I?D ???C? THIRD something?

37D - Party defeats Greens (4, 6) ?I?E ?I?H?S

37 Side Dishes, greens. Side, party +Dishes, defeats.

33 Third Place, bronze - an anagram.

50 Consigliere, adviser. I think it's an anagram (unorthodox) of 'religions' and CE, but I didn't do it!

Thanks Malone!

Myla, glad I could help.

29a "I'll be seized by hallucinatory experiences after knocking back mescal?" (6)

S?I?I? assuming 23d's S is right SPIRIT would fit but not sure of parsing

Any help much appreciated

Yes ... spirits .
I in trips (hallucinatory experiences) reversed

29, the hallucinatory experiences are "trips", reversed around "I".
Spirit.