The DT has the songs "Fair Nottamun Town" and Paddy Backwards. Thread 13671 discusses this song: Nottamun Town .
Vance Randolph, "Roll Me In Your Arms," pp. 302-305, has three versions of "Nottingham Fair" aka "Nottamon Fair" of which one is fairly complete. These "lying songs" (of Irish type?) concern a visit to the fair by a young man and the improbable sights and happenings there.
They seem rather distantly related to the hunting song. Lying songs have always been popular.

A short one from Randolph:

Nottingham Fair

As I was a-walkin' to Nottingham Fair,
I seen a fair damsel all on a gray mare,
With her ass painted blue and a bull on her back,
A bundle of fodder was stuck in her crack.

And there was the King and a company more,
A-riding on horseback all walkin' before,
A stark naked drummer a-beating the drum,
With his heels in his ass-hole before them did run.

The mare throwed the damsel right off in the ditch,
So I out with my old doogey an' mounted the bitch,
She drawed back her foot an' kicked me in the shin,
Before I got on I was off her ag'in.

It is stretching to relate these songs.