The most common and most likely answer to this crossword clue is the 10 letter word EPISTERNUM
E
P
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S
T
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M
Buddy explains!
The straight part
We know that "a bit of chest" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case a bit of chest = episternum.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 EPISTERNUM mean?
EPISTERNUM noun- The upper segment of the sternum
- The anterior part of the pleuron of an insect.