Controls art gallery intended, we hear, for restoration (13)
I believe the answer is:
reinstatement
'for restoration' is the definition.
(reinstatement is a kind of restoration)
'controls art gallery intended we hear' is the wordplay.
'controls' becomes 'reins' (take the reins).
'art gallery' becomes 'tate' (Tate Modern, Tate Britain etc).
'intended' becomes 'meant' (to mean something is to intend it).
'we hear' shows a homophone (sound like).
'meant' sounds like 'ment'.
'reins'+'tate'+'ment'='REINSTATEMENT'
(Other definitions for reinstatement that I've seen before include "Restoring" , "Rehabilitation" , "Restoration to a previous position" .)