The most common and most likely answer to this crossword clue is the 4 letter word SOOT
S
O
O
T
Buddy explains!
The straight part
We know that "Something stuck down the chimney" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case something stuck down the chimney = soot.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 SOOT mean?
SOOT noun- Fine black or dull brown particles of amorphous carbon and tar, produced by the incomplete combustion of coal, oil etc.
- To cover or dress with soot.