The most common and most likely answer to this crossword clue is the 12 plus 1 plus 1 letter word PEREGRINATOR - A
I've seen this crossword clue in The Times
P
E
R
E
G
R
I
N
A
T
O
R
-
A
Buddy explains!
The straight part
We know that "person on odyssey" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case person on odyssey = peregrinator - a.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.