This appears to be a family that was living in Owsley County, Kentucky in the 1850 U.S. Census. The death ceritifcate for son William, (born in North Carolina about 1828) gives his mother's maiden name as "Carter." There are two David Robinson families in Buncombe County on the 1830 census. I don't find any in 1820. The family apparently moved to Kentucky about 1840. Find-a-grave says David's wife was a woman named Nancy Runyons who married a man named John Carter in 1823, but there is no record of a Nancy Runyons marrying David Robinson.. I don't find ANY marriage records for David Robinson.
There is nothing in this family to suggest a Cherokee connection. Buncombe County was not in the Cherokee Nation There is no one named Robinson on the 1835 Cherokee Census or the 1851/52 rolls of Cherokee East of the Mississippi. White men who married Cherokee women lived in the Cherokee Nation, not the other way around.