The most common and most likely answer to this crossword clue is the 11 letter word MAN-AND-DOG
I've seen this crossword clue in The Guardian
M
A
N
-
A
N
D
-
D
O
G
Buddy explains!
The straight part
We know that "with minimum staff" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case with minimum staff = man-and-dog.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.