The most common and most likely answer to this crossword clue is the 7 plus 4 letter word KIT-CAT CLUB
I've seen this crossword clue in The Guardian
K
I
T
-
C
A
T
C
L
U
B
Buddy explains!
The straight part
We know that "Political coterie" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case political coterie = kit-cat club.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.