The most common and most likely answer to this crossword clue is the 9 letter word RULERSHIP
I've seen this crossword clue in Independent
R
U
L
E
R
S
H
I
P
More clues leading to the result RULERSHIP
Buddy explains!

The straight part
We know that "monarch’s office" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case monarch’s office = rulership.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.