Record held by Peter for one branch of the family (4)
I believe the answer is:
sept
'family' is the definition.
('sept' can be a synonym of 'family')
'record held by peter' is the wordplay.
'record' becomes 'ep'.
'held by' means one lot of letters goes inside another.
'peter' becomes 'st' ('St Peter' can be a synonym of 'peter').
'ep' going into 'st' is 'SEPT'.
'for one branch of the' is the link.
I am not very happy about this link. Some or all of it may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for sept that I've seen before include "Branch of Irish clan" , "Thirty days" , "Irish family group" , "Short month" , "Month (abbreviation); clan subdivision (in Ireland)" .)