Small nucleus with these distributed record of game (10)
I believe the answer is:
scoresheet
'record of game' is the definition.
(the record of scores in a game)
'small nucleus with these distributed' is the wordplay.
'small' becomes 's' (abbreviation - e.g. clothes size).
'nucleus' becomes 'core' (synonyms).
'with' says to put letters next to each other.
'distributed' indicates anagramming the letters.
'these' anagrammed gives 'sheet'.
's'+'core'+'sheet'='SCORESHEET'
(Other definitions for scoresheet that I've seen before include "goal to get on this?" , "Record of players' performances" .)