The most common and most likely answer to this crossword clue is the 8 letter word EXORCISM
I've seen this crossword clue in The Guardian
E
X
O
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C
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M
Buddy explains!
The straight part
We know that "Minister’s job can be /d/ispiriting" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case minister’s job can be /d/ispiriting = exorcism.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 EXORCISM mean?
EXORCISM noun- The ritual act of driving out evil spirits from persons, places or things who are possessed by them.