“Spital” is a contraction of “Hospital”. This is probably the Spital house on the West End Road (B6531) out of Hexham, now the Golf Club. There are some photos of this in our Photo Archive: https://www.rogerb129.sg-host.com//historic-hexham/photograph-archive/photo-archive-choices/hexham-choices/buildings-choices/spital-gallery/
Some further information:
1811. Now an elegant, handsome house, lately built by James Gibson Kirsopp, Esq. the
present possessor: it stands upon an elevated ground, at a hundred and fifty yards distance
from the Tyne on the south bank, and commands a fine prospect for ten or twelve miles
eastward down the vale, as also a good way up the North and West Tynes, being a little
below the conflux of these branches. The Spittal formerly belonged to Hexham church, and
was called St. Giles’s Hospital, but was dismembered by some means not now known,
about the time of Henry VIII. History of Northumberland, Vol. II, Mackenxzie and Dent, 1811, p.346.