Yearning to break most of finest possessions (10)
I believe the answer is:
belongings
'possessions' is the definition.
(belonging is a kind of possession)
'yearning to break most of finest' is the wordplay.
'yearning' becomes 'longing' ('long' can be a synonym of 'yearn').
'to break' means one lot of letters goes inside another.
'most of' means to remove the last letter.
'finest' becomes 'best' (similar in meaning).
'best' with its final letter taken away is 'bes'.
'longing' inserted inside 'bes' is 'BELONGINGS'.
(Other definitions for belongings that I've seen before include "property" , "stuff" , "Personal effects" .)