The word "Neenyah" comes from the Spanish "niña" meaning (little) girl, spelled so English-speaking people can pronounce it. To get to pronounce the Spanish word "mi", it is spelled "me" in the title. So the title in Spanish would be "Mi niña."
The Fletcher Henderson band has a recording of the tune.
More than a Spanish tinge, I hear a mixed Spanish-Oriental oriental tinge in the melody.
Another version, this one by Julian Fuhs.
Both of these versions swing a lot more than the Henri Gendron version.