You can hear this version at the Internet Archive:

THE MAN THAT COMES AROUND
(Lair/Tucker)
As recorded by Tommy Tucker Time, with vocal by Kerwin Somerville, on Vocalion 5199, 1939.

CHORUS: [Now,] there's a man that comes to our house ev'ry single day.
Papa comes home and the man goes away.
Oh, Papa does the work and Mama gets the pay,
And the man comes around when Papa goes away.

1. Now, the man that comes to our house, he's very nice.
He comes in the summertime to bring my mama ice,
But the little teeny weenie piece soon melts away,
And he has to come back again later in the day. CHORUS

2. The man that comes to our house, he comes to take the trash
In a little white jacket and a little black mustache.
It's all very strange, but it always seems to me
He's a little more familiar than the trash man ought to be. CHORUS

3. The man that comes to our house, he comes to bring the milk.
He walks right in the kitchen and he talks as smooth as silk.
I always have to hold his horsie out by the gate,
'Cause he always stays so awful long the horse don't want to wait. CHORUS

4. Now when I grow up, I'm never gonna be
A plumber or a carpenter, no sirree!
I'll never be a doctor with an office downtown.
No, I'd rather be just the man that comes around. CHORUS

TAG: Whoa-ho-ho-ho! I'd rather be just the man that comes around!

[Another recording by the same band and vocalist appears on Columbia 37082, from 1941. The recording quality is better, and the vocal is a bit more expressive, but the lyrics are practically identical.]