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