The most common and most likely answer to this crossword clue is the 4 plus 2 plus 1 plus 6 letter word HANG BY A THREAD
I've seen this crossword clue in Financial Times
H
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G
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Buddy explains!
The straight part
We know that "Look precarious/as did Damocles’ sword?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case look precarious/as did damocles’ sword? = hang by a thread.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.