The most common and most likely answer to this crossword clue is the 4 letter word LUNG
I've seen this crossword clue in The Times
L
U
N
G
More clues leading to the result LUNG
Buddy explains!

The straight part
We know that "a breather" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case a breather = lung.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 LUNG mean?
LUNG noun- A biological organ of vertebrates that controls breathing and oxygenates the blood.
- (plural) Capacity for exercise or exertion; breath.
- That which supplies oxygen or fresh air, such as trees, parklands, forest, etc., to a place.