The most common and most likely answer to this crossword clue is the 8 letter word NOSTRILS
I've seen this crossword clue in The Guardian
N
O
S
T
R
I
L
S
More clues leading to the result NOSTRILS
Buddy explains!
The straight part
We know that "Breathing spaces" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case breathing spaces = nostrils.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 NOSTRILS mean?
NOSTRILS noun- Either of the two orifices located on the nose (or on the beak of a bird); used as a passage for air and other gases to travel the nasal passages.