The most common and most likely answer to this crossword clue is the 7 letter word SATRAPS
I've seen this crossword clue in The Daily Telegraph
S
A
T
R
A
P
S
Buddy explains!

The straight part
We know that "Governors" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case governors = satraps.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 SATRAPS mean?
SATRAPS noun- A governor of a Persian province.
- A subordinate ruler.