Twenty losing head and heart (4)
I believe the answer is:
core
'heart' is the definition.
('core' can be a synonym of 'heart')
'twenty losing head' is the wordplay.
'twenty' becomes 'score' ('score' can be a synonym of 'twenty').
'losing head' means to remove the first letter.
'score' with its first letter taken away is 'CORE'.
'and' acts as a link.
(Other definitions for core that I've seen before include "Heart of the matter" , "Central part of an apple" , "Central or most important part of something" , "Central part of the Earth" , "Centre, heart" .)