Normally I can struggle to get one answer this week I nearly finished it!
1d. Supernatural being priest exorcised (6). S?R?T? Thinking it is sprite
20d. Man from ancient city calling for immediate attention (6). ?R?E?T all I can see is urgent.
27a. Very busy with calls, daughter in other resort (3,3) ?E? H?T Thinking red hot but not sure.
18d. Shelters for animals, well groomed, reared round outskirts of Nuneaton (7). H?N?E?S thinking henpens but that is only because it fits.