The most common and most likely answer to this crossword clue is the 2 plus 4 plus 9 letter word ON GOOD AUTHORITY
I've seen this crossword clue in The Guardian
O
N
G
O
O
D
A
U
T
H
O
R
I
T
Y
Buddy explains!
The straight part
We know that "as one may well believe" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case as one may well believe = on good authority.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.