The most common and most likely answer to this crossword clue is the 7 plus 4 letter word PROJECT FEAR
I've seen this crossword clue in Financial Times
P
R
O
J
E
C
T
F
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A
R
Buddy explains!
The straight part
We know that "sending out a negative message" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case sending out a negative message = project fear.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.