The most common and most likely answer to this crossword clue is the 3 plus 8 plus 8 letter word THE PILGRIMS PROGRESS
I've seen this crossword clue in The Times
T
H
E
P
I
L
G
R
I
M
S
P
R
O
G
R
E
S
S
Buddy explains!
The straight part
We know that "Book/journey on the Mayflower?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case book/journey on the mayflower? = the pilgrims progress.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.