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Dear Ms. Paddock,
Thank you for posting your request on History Hub!
The Compiled Military Service Records (CMSR) and the pension files may include information pertaining to Francis Cyprien Kiern. For access to and/or copies of these series, please contact the National Archives at Washington, DC (RDT1) via email at email@example.com. These records are also available on ancestry.com and/or familysearch.org.
On the New York State Military Museum and Veterans Research Center, we located a unit history of Co C. 20 Infantry Regiment, 7th Corps New York. This may be the same unit listed on his gravestone.
We hope this information is helpful. Best of luck with your family research!