The warmth gets right into the centre (5)
I believe the answer is:
heart
'the centre' is the definition.
(synonyms)
'the warmth gets right' is the wordplay.
'the warmth' becomes 'heat' ('heat' can be a synonym of 'warmth').
'gets' is an insertion indicator (gets can mean captures or absorbs).
'right' becomes 'r' (common abbreviation).
'heat' enclosing 'r' is 'HEART'.
'into' is the link.
(Other definitions for heart that I've seen before include "Muscular organ for blood circulation" , "bottle" , "Cardiac organ" , "Core; organ" , "Middle" .)