Demesne has eastern and southern gallery (6)
I believe the answer is:
estate
'demesne' is the definition.
(I know that demesne can be written as estate)
'eastern and southern gallery' is the wordplay.
'eastern' becomes 'e'.
'and' means one lot of letters go next to another.
'southern' becomes 's' (abbreviation).
'gallery' becomes 'tate' (Tate Modern, Tate Britain etc).
'e'+'s'+'tate'='ESTATE'
'has' is the link.
(Other definitions for estate that I've seen before include "Houses - car" , "Family land" , "The entire property of a deceased person" , "Property - may go with rank" , "Everything one owns -- tea set (anag)" .)