As far as I can tell, there is no connection between this song and the famous instrumental jazz standard written as a piano rag in 1914 by Lew Pollack, and used as a theme by Jackie Gleason. I don't even know what the following song sounds like, being unable to read music myself, and having failed to find a recording. (Any search is likely to lead to the aforementioned Pollack tune.)

From the sheet music at Baylor University:


THAT'S A PLENTY
Words by Henry S. Creamer, music by Bert A. Williams, ©1909.

1. 'Twas in a hen-coop one dark night, I stopped to rest myself,
And to my great surprise, I spied some pullets on a shelf.
I tried to shake hands with a hen when some buckshot passed my head.
I dropped my bag and turned around, then to the gen'man said:

CHORUS 1: "Mister, mister, oh, listen like a friend.
No use to argue; just let the matter end.
Excuse me, boys, if wrong I've done.
There ain't no use to shoot that gun.
Just move that much so I can run.
Thank you! That's a plenty."

2. I felt a cramp one morning; to the doctor I ran quick.
He said I had the epizootic; man, I sure was sick!
He said that I should bathe in cold water each morning twice,
But I told him when he told me to fill the tub with ice:

CHORUS 2: "Mister, mister, oh, listen like a friend.
No use to argue; we'll let that treatment end.
Remember I'm no Eskimo.
I do not care for ice or snow.
I'll take one bath in July though.
Just one; that's a plenty."

3. A man gave me a two-bit note to guide him through the woods,
And as we strolled along, a bear jumped out; it was the goods!
The man skiddooed; I grabbed that bear; it made an awful wail.
The man yelled: "Stop" but I said: "No; I'm goin' to hold this tail.

CHORUS 3: "Mister, mister, oh, listen like a friend.
No use to tell me to hold the other end.
I see I've got to hold this bear.
I just leave hold him here as there.
As long as his tail's got one hair
That long; that's a plenty."