Terribly fond of holding son, goes to sleep (4,3)
I believe the answer is:
nods off
'goes to sleep' is the definition.
('nod off' can be a synonym of 'go to sleep')
'terribly fond of holding son' is the wordplay.
'terribly' is an anagram indicator.
'holding' means one lot of letters goes inside another.
'son' becomes 's'.
'fond'+'of'='fondof'
'fondof' with letters rearranged gives 'nodoff'.
'nodoff' placed around 's' is 'NODS OFF'.
(Other definitions for nods off that I've seen before include "Falls asleep" , "Dozes" , "loses consciousness" , "Goes out" .)