The most common and most likely answer to this crossword clue is the 6 letter word TONSIL
I've seen this crossword clue in The Times
T
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More clues leading to the result TONSIL
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The straight part
We know that "Often unwanted tissue" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case often unwanted tissue = tonsil.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 TONSIL mean?
TONSIL noun- Either of a pair of small masses of lymphoid tissue that lie on each side of the throat and that help protect the body against infection; palatine tonsil.
- Any of various small masses of lymphoid tissues, including palatine tonsils, adenoids and lingual tonsils.