The most common and most likely answer to this crossword clue is the 10 letter word ARCHDEACON
I've seen this crossword clue in Financial Times
A
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C
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More clues leading to the result ARCHDEACON
- Bow to extremely defensive prisoner gripping a clergyman
- Cleric’s strange charade over swindle
- Triumphal building, notion originally dismissed by right-winger, senior minister
- Church official‘s mischievous, having area coned off outside
- A Catholic had once represented senior cleric
- Church dignitary maintaining firm support at first for another
- Bridge — one cad played, church official
- Company taken in by cunning academic or member of clergy
- Company taken in by cunning university official and cleric
- Bridge — after shuffle, ace is held by academic or cleric
Buddy explains!

The straight part
We know that "senior cleric" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case senior cleric = archdeacon.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 ARCHDEACON mean?
ARCHDEACON noun- In the Anglican and Eastern Orthodox systems, a senior administrative official in a diocese, just under the bishop, often in charge of an archdeaconry. As a title, it can be filled by either a deacon or priest.