Many a mug means a great deal! (4)
I believe the answer is:
mass
'a great deal' is the definition.
(I know that slew can be written as mass)
'many a mug' is the wordplay.
'many' becomes 'm' (Roman numeral for a thousand).
'a mug' becomes 'ass' (I've seen this before).
'm'+'ass'='MASS'
'means' is the link.
(Other definitions for mass that I've seen before include "congregate for > service" , "load" , "congregation for > service" , "Body of matter" , "Church service - crowd" .)