The most common and most likely answer to this crossword clue is the 3 plus 7 plus 6 letter word THE FASHION POLICE
I've seen this crossword clue in The Guardian
T
H
E
F
A
S
H
I
O
N
P
O
L
I
C
E
Buddy explains!
The straight part
We know that "If cheap clothes on, I suspect, cops primarily sent out" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case if cheap clothes on, i suspect, cops primarily sent out = the fashion police.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.