Swear to give a cry of pain with a heavier heart (3)
I believe the answer is:
vow
'swear' is the definition.
(I've seen this before)
'a cry of pain with a heavier heart' is the wordplay.
'a cry of pain' becomes 'ow' (exclamation of pain).
'with' is a charade indicator (letters next to each other).
'heart' indicates the central letters.
The central letter of 'heavier' is 'v'.
'ow' after 'v' is 'VOW'.
'to give' is the link.
(Other definitions for vow that I've seen before include "Pledge, take oath" , "Plight" , "Solemn pledge" , "Oath or solemn promise" , "Binding promise" .)