In tears, beg chap desperately for some respite (9,5)
I believe the answer is:
'some respite' is the definition.
(breathing space is a kind of respite)
'in tears beg chap desperately' is the wordplay.
'desperately' indicates an anagram.
'intearsbegchap' is an anagram of 'BREATHING-SPACE'.
'for' acts as a link.
(Other definitions for breathing space that I've seen before include "chance to relax", "room", "might give one an opportunity to relax", "Brief respite", "relief".)