The most common and most likely answer to this crossword clue is the 5 letter word SYNOD
I've seen this crossword clue in Independent
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More clues leading to the result SYNOD
- Clerical group mired in controversy, no doubt
- Yankee stops fellows travelling up for meeting
- Academics upset about conclusion of priestly council
- Church group’s crime not even reported
- Tips off secretary before backing academic council
- Assembly ends on Saturday with agreement
- Yankee stops lecturers travelling up for meeting
- Diocesan council, say, ignoring a show of consent
- Vacant seminary gets expression of approval from church council
- Where you find some heresy? No; dogma!
Buddy explains!

The straight part
We know that "church council" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case church council = synod.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does SYNOD mean?
SYNOD noun- An ecclesiastic council or meeting to consult on church matters.
- An administrative division of churches, either the entire denomination, as in the Lutheran Church–Missouri Synod, or a mid-level division (middle judicatory, district) as in the Evangelical Lutheran Church in America
- An assembly or council having civil authority; a legislative body.
- A conjunction of two or more of the heavenly bodies.