I'm not sure it's the same song, but Wingy Manone & His Orchestra recorded one called THE MOSQUITO SONG (WHERE DO MOSQUITOES GO IN THE WINTER TIME?), Bluebird 10773, 1940. It's on 2 albums: Various-artists, "Laughter from the Hip: 24 Jazz Comedy Classics," and Wingy Manone & His Orchestra, "1939-1940."

Here's what I was able to transcribe from a couple of sound samples:

[Intro, spoken:]
—Say, Bub? Boy, you know about them bird-big mosquitoes they got over there in Jersey?
—You mean those big gallow-whoppers?
—Yes!
—Those kind that bite you when you're not even lookin'?
—Yes! I'm gonna tell you what I want to know:

[Sings:]
[CHORUS?] Now what I want to know
Is where mosquitoes go
When the Jersey breezes blow.
. . .
[VERSE?] …for quite a long, long time.
We'll never rest until they go.
[BRIDGE?] Some folks say they go to the woodland.
Some folks say they go to the well.
They way they bite and itch me, (Ah!)
I hope they go to— (No!).
[CHORUS?] What I want to know
Is where mosquitoes go
When the Jersey breezes blow.