The most common and most likely answer to this crossword clue is the 10 letter word ARCHPRIEST
I've seen this crossword clue in The Times
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Buddy explains!
The straight part
We know that "old bishop's assistant" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case old bishop's assistant = archpriest.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 ARCHPRIEST mean?
ARCHPRIEST noun- (Eastern Orthodox Church) The highest rank given to a married priest.
- (Roman Catholic Church) An honorific title applied to a priest who has a specific function.