Whitehall edifice not cheap to refurbish (8)
I believe the answer is:
cenotaph
'whitehall edifice' is the definition.
(memorial located in Whitehall)
'not cheap to refurbish' is the wordplay.
'to refurbish' indicates an anagram (I've seen 'refurbished' mean this).
'not'+'cheap'='notcheap'
'notcheap' with letters rearranged gives 'CENOTAPH'.
(Other definitions for cenotaph that I've seen before include "Empty tomb" , "Whitehall monument" , "War memorial" , "Monument to those killed in war" , "Monument to the war dead" .)