The most common and most likely answer to this crossword clue is the 4 plus 4 letter word PULL RANK
I've seen this crossword clue in The Daily Telegraph
P
U
L
L
R
A
N
K
More clues leading to the result PULL RANK
Buddy explains!

The straight part
We know that "Exert one’s authority" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case exert one’s authority = pull rank.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.