Split or part after hugging a moment (6)
I believe the answer is:
'split' is the definition.
(I've seen this before)
'part after hugging a moment' is the wordplay.
'part' becomes 'bit' (both mean a piece of something).
'after hugging' is an insertion indicator.
'a moment' becomes 'sec' (short for second).
'bit' placed around 'sec' is 'BISECT'.
'or' is the link.
(Other definitions for bisect that I've seen before include "Cut in two equal parts" , "Cut equally in two" , "equally split" , "Split exactly in two" , "create schism?" .)