The most common and most likely answer to this crossword clue is the 6 letter word HERMAN
I've seen this crossword clue in Independent
H
E
R
M
A
N
Buddy explains!
The straight part
We know that "could be Woody (Woody Herman, US jazz musician)" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case could be woody (woody herman, us jazz musician) = herman.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.