Rob,

With only this information, just name and county, to go on this is a possibility:

John H. Cunnigham
PVT
Co. B, 4th Tenn. Cav. Battalion

4th Battalion, Tennessee Cavalry (Branner's)
4th Cavalry Battalion [also called 1st or 2nd Battalion] was organized in August, 1861, with six companies. Its members were from the counties of Knox, Hamilton, Monroe, Union, Claiborne, Hawkins, Bledsoe, and Jefferson. In September the unit contained 490 effectives and confronted the Federals in Kentucky. During May, 1862, it merged into the 2nd (Ashby's) Tennessee Cavalry Regiment. Lieutenant Colonel Benjamin M. Branner and Major John M. Bridgman were in command.

John H. Cunningham
PVT
Co. G, 2nd Tenn. Cav. Regt.

2nd Regiment, Tennessee Cavalry (Ashby's)
2nd (Ashby's) Cavalry Regiment was formed by consolidating the 4th and 5th Battalions of Tennessee Cavalry in May, 1862. The men were from the counties of Hamilton, Monroe, Knox, Union, Claiborne, Hawkins, Jefferson, Bledsoe, Bradley, Polk, Hancock, Blount, and Sullivan. It was assigned to Wharton's, Pegram's, J.S. Scott's, Davidson's, Humes', and H.M. Ashby's Brigade. The regiment participated in the conflicts at Cumberland Gap, Murfreesborough, and Chickamauga, then was involved in the Atlanta Campaign. Later it skirmished in Alabama, aided in the defense of Savannah, and took part in the campaign of the Carolinas. It lost 6 killed, 8 wounded, and 19 missing at Cumberland Gap, and reported 19 casualties at Chickamauga. Only a few surrendered with the Army of Tennessee. The field officers were Colonel Henry M. Ashby, Lieutenant Colonels Henry C. Gillespie and John H. Kuhn, and Majors Pharaoh A. Cobb and William M. Smith.

As you see these units are one and the same. These two, John H. Cunningham, are probably the same man.

Len