The most common and most likely answer to this crossword clue is the 8 letter word NEUROSES
I've seen this crossword clue in The Daily Telegraph
N
E
U
R
O
S
E
S
Buddy explains!
The straight part
We know that "one’s hang-ups" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case one’s hang-ups = neuroses.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 NEUROSES mean?
NEUROSES noun- A mental disorder, less severe than psychosis, marked by anxiety or fear which differ from normal measures by their intensity, which disorder results from a failure to compromise or properly adjust during the developmental stages of life, between normal human instinctual impulses and the demands of human society.