The most common and most likely answer to this crossword clue is the 9 letter word INCUMBENT
I've seen this crossword clue in The Guardian
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More clues leading to the result INCUMBENT
- Priest maybe is necessary
- Post holder’s description of US bodies — I hesitate to say: corrupt
- Speaker's earnings corrupt post-holder
- Office holder, one new to money, secures university degree
- Money received in report by corrupt priest?
- Up to official
- Office-holder and smart copper, crooked, pocketing million
- At home with dishonest office-holder
- Living man has a duty
- Obligatory holder of post
Buddy explains!

The straight part
We know that "priest" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case priest = incumbent.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 INCUMBENT mean?
INCUMBENT noun- The current holder of an office, such as ecclesiastical benefice or an elected office.
- A holder of a position as supplier to a market or market segment that allows the holder to earn above-normal profits.
- Imposed on someone as an obligation, especially due to one's office.
- Lying; resting; reclining; recumbent.
- Prevalent, prevailing, predominant.
- Resting on something else; in botany, said of anthers when lying on the inner side of the filament, or of cotyledons when the radicle lies against the back of one of them.
- Bent downwards so that the ends touch, or rest on, something else.
- Being the current holder of an office or a title.