The most common and most likely answer to this crossword clue is the 7 letter word EXHALED
I've seen this crossword clue in The Sunday Telegraph
E
X
H
A
L
E
D
More clues leading to the result EXHALED
Buddy explains!

The straight part
We know that "Gave off" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case gave off = exhaled.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 EXHALED mean?
EXHALED verb- To expel air from the lungs through the nose or mouth by action of the diaphragm, to breathe out.
- To expel (something, such as tobacco smoke) from the lungs by action of the diaphragm.
- To pass off in the form of vapour; to emerge.
- To emit (a vapour, an odour, etc.).
- To draw out; to cause to be emitted in vapour.