Testing your picture upload.
This is a pic of an old mill on Carriacou. Well, the remains of it.

Nice picture.
Were these ills used for the sugar cane?

These mills were used to grind sugar cane for making molasses, rum, and raw sugar, the British built windwills like this one at Belair.

The havoc wrought upon Carriacou by the introduction of a sugar economy has been called the "overdevelopment" of Carriacou. And although sugar provided much of the wealth required for the development of capitalism and the Industrial Revolution, people in places like Carriacou continue to live with the effects of this overdevelopment, which include destruction of natural ecosystems leading to the lowering of water tables and erosion of arable soils, culturally manufactured dependence on the former colonial powers, and overpopulation.

this building is behind the hospital. I was told that it was a lime kiln

I think both of you are right.
Probably it was first used for the sugar, then later for the limes which were introduced to Carriacou when the sugar was not cultivated on the island anymore due to lowered sugar prices, land erosion and abolition.