Hi Scotsmum, Yes, this would have been Cleland Hospital, previously the Combination Poorhouse. As far as I know it's still there, although it's been a couple of years since I was over that way.
It would have been called the "County" hospital as opposed to, for example, the one in Motherwell which was known as the Burgh Hospital. Anyplace that wasn't in a burgh (the burghs of Lanarkshire being Hamilton, Motherwell & Wishaw, Lanark, Airdrie & Coatbridge) was considered to be in the County, for example, Cleland, Bellshill, Shotts, Carluke, Larkhall (I think) and other small villages.
There was a "County Hospital" on the outskirts of Motherwell but that was a Fever Hospital that served the whole county of Lanark.