The most common and most likely answer to this crossword clue is the 10 letter word COUNCILMEN
I've seen this crossword clue in The Guardian
C
O
U
N
C
I
L
M
E
N
Buddy explains!
The straight part
We know that "chaps running local government in America" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case chaps running local government in america = councilmen.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 COUNCILMEN mean?
COUNCILMEN noun- A male member of a council, especially a city council.