Support a pillow (7)
I believe the answer is:
bolster
This clue is a double definition.
'support' is the first definition.
('bolster' can be a synonym of 'support')
'a pillow' is the second definition.
(bolster is a kind of pillow)
(Other definitions for bolster that I've seen before include "Lobster (anag.)" , "Extended pillow" , "Shore up" , "support for head after retiring" , "Long thick pillow" .)