The most common and most likely answer to this crossword clue is the 2 plus 3 plus 3 plus 6 letter word ON THE OFF CHANCE
I've seen this crossword clue in The Daily Telegraph
O
N
T
H
E
O
F
F
C
H
A
N
C
E
More clues leading to the result ON THE OFF CHANCE
Buddy explains!
The straight part
We know that "In hope rather than expectation" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case in hope rather than expectation = on the off chance.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.