Mum’s letter means a lot (4)
I believe the answer is:
mass
'a lot' is the definition.
(synonyms)
'mum's letter' is the wordplay.
'mum's' becomes 'mas' ('ma' can be a synonym of 'mum').
'letter' becomes 's' ().
'mas'+'s'='MASS'
'means' is the link.
(Other definitions for mass that I've seen before include "Quantity; religious service" , "Large amount, body" , "Body of matter" , "Accumulation" , "Service; horde" .)