The most common and most likely answer to this crossword clue is the 6 letter word BLOWSY
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More clues leading to the result BLOWSY
Buddy explains!

The straight part
We know that "Coarse" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case coarse = blowsy.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 BLOWSY mean?
BLOWSY adjective- Having a reddish, coarse complexion, especially with a pudgy face.
- (chiefly of a woman's hair or dress) Slovenly or unkempt, in the manner of a beggar or slattern.
- Unrefined, countrified.