The most common and most likely answer to this crossword clue is the 8 letter word DEEP-SET
I've seen this crossword clue in Independent
D
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P
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More clues leading to the result DEEP-SET
Buddy explains!

The straight part
We know that "embedded firmly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case embedded firmly = deep-set.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 DEEP-SET mean?
DEEP-SET adjective- Set deeply below a surface, as on a face.