A 'jenny' is a term for a female donkey. The tune is related to "Paddy on the Turnpike" (in the Irish section). Alan Jabbour sees the antecedent of the tune, at least in America, as "Colonel Crocket" which appears in George Knauff's Virginia Reels, vol. III (Baltimore, 1839). The melody also has strong similarities to a tune called "The Route".
It has been recorded by New Lost City Ramblers on String Band Instrumentals (1964), Bob Carlin & John Hartford and others.