Expression of relief from hardly any by the sound of it (4)
I believe the answer is:
phew
'expression of relief' is the definition.
(sound made when relieved)
'hardly any by the sound of it' is the wordplay.
'hardly any' becomes 'few' (similar in meaning).
'by the sound of it' indicates a 'sounds like' (homophone) clue.
'few' is a homophone of 'PHEW'.
'from' is the link.
(Other definitions for phew that I've seen before include "Sound relieved" , "Recover unharmed" , "Expression of relief" , "Gosh, that's a relief!" , "Good gracious, its hot!" .)