Gallery that has heartless taste (4)
I believe the answer is:
tate
'gallery' is the definition.
(Tate Modern, Tate Britain etc)
'heartless taste' is the wordplay.
'heartless' says to hollow out the word (remove centre letters).
'taste' with its centre removed is 'TATE'.
'that has' acts as a link.
(Other definitions for tate that I've seen before include "Art gallery benefactor" , "Art gallery endowed by a sugar merchant" , "Well-known London art gallery" , "Lord Sugar" , "Sugar merchant and art gallery philanthropist, founder, d. 1899" .)