Tennesseans in the Civil War, records a John H. Cunningham as in Co. G, 2nd Tenn. Cav.

Also, this same publication records Co. G as from Monroe County.

I did an Ancestry.com search and found in the 1860 census a 20 year old John H. Cunningham, farmer, in the household of Sarah Cunningham, age 48, also listed as a farmer.

This was in the 6th Civil District, Sweetwater Post Office,
Monroe County, Tenn.

This could be your man.

Good Hunting,

George Martin