The old record could make one cry (4)
I believe the answer is:
yelp
'one cry' is the definition.
(yelp is a kind of cry)
'the old record' is the wordplay.
'the old' becomes 'ye'.
'record' becomes 'lp' (LP is an example).
'ye'+'lp'='YELP'
'could make' is the link.
(Other definitions for yelp that I've seen before include "Short sharp cry (of such as a dog)" , "Barking" , "Cry of pain" , "Express discomfort" , "(Eg, dog's) short cry of alarm" .)