The most common and most likely answer to this crossword clue is the 6 letter word PILATE
I've seen this crossword clue in Financial Times
P
I
L
A
T
E
More clues leading to the result PILATE
- Very religious old procurator of Judaea
- Governor in India having dish brought round
- Old governor, good and dead
- Roman governor gets a lot of exercise
- With exercise unfinished, he washed his hands
- Trial being heard — man presiding at it?
- Devout former judge and governor
- Brigand swaps sides to be Roman governor
- Governor‘s exercise regime cut short
- Self-righteous, missing an appointment? He ceremoniously disclaimed responsibility
Buddy explains!

The straight part
We know that "Governor" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case governor = pilate.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.