The most common and most likely answer to this crossword clue is the 10 letter word PRIESTHOOD
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More clues leading to the result PRIESTHOOD
Buddy explains!

The straight part
We know that "Ministry" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case ministry = priesthood.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 PRIESTHOOD mean?
PRIESTHOOD noun- The role or office of a priest.
- Priests as a group; the clergy.
- Authority to act in the name of God.