I don't have record of a Richard Harness, but there is a Thomas Harness listed on the rolls of Co. G, 3rd Arkansas Cavalry, U.S. Volunteers.
He enlisted in Conway County on October 28, 1863, for three years. The descriptive roll describes him as a farmer, age 36, born in Wayne County, Tennessee, eyes hazel, hair light, complexion fair, height 6'1". He applied for a U.S. veteran's pension in 1880.
If this sounds like the veteran you're hunting for, you can order a copy of his service and pension records from the Military Records Research site linked to this message board -- http://history-sites.com/research/index.html