The most common and most likely answer to this crossword clue is the 5 letter word HOPE*
I've seen this crossword clue in The Guardian
H
O
P
E
*
Buddy explains!
The straight part
We know that "Have some expectation of success" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case have some expectation of success = hope*.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.
What does HOPE* mean?
HOPE* verb- To want something to happen, with a sense of expectation that it might.
- To be optimistic; be full of hope; have hopes.
- To place confidence; to trust with confident expectation of good; usually followed by in.
- To wish.
- The feeling of trust, confidence, belief or expectation that something wished for can or will happen.
- The actual thing wished for.
- A person or thing that is a source of hope.
- The virtuous desire for future good.
- A hollow; a valley, especially the upper end of a narrow mountain valley when it is nearly encircled by smooth, green slopes; a comb.
- A sloping plain between mountain ridges.
- A small bay; an inlet; a haven.