I think the mystery section is a reference to a specific brand of cigarettes: https://en.wikipedia.org/wiki/Fatima_(cigarette)
Here’s my best transcription so far:

Mama, Mama, said mister absently
You seem to worry me.
Why all this mystery?
Mama, Mama, the way you talk these days,
I have to guess your ways.
You're like the crossword craze.

Crossword Mama, you puzzle me.
But Papa's gonna figure you out.
You call me honey an’ that means bee
Looks like I'll be stung no doubt.

You go out calling for an hour or two.
I don't know where or when or how or who.
Crossword Mama, you puzzle me,
But Papa's gonna figure you out.

Your Papa's gonna crossword you right now.
You better get your answers right.
I heard you mention "butcher"—that means "meat"!
Who you gonna "meet" tonight?

Every time I phone, you say that you're alone,
Working on a crossword phrase.
Every time I call, you run across the hall.
You cross me fifty-seven ways.

Washington, he crossed the Delaware.
Columbus crossed the ocean blue.
If there's any more crossing to be done,
Papa's gonna double-cross you.

Dee-dee-dee [etc.]

Mama, Mama, you're giving me the blues.
You're acting like bad news,
So watch your Ps and Qs.
Mama, Mama, I've got an ax to grind.
There’s lots of things I find
That’s running through my mind.

Crossword Mama, you puzzle me,
But Papa's gonna figure you out.
Every key to a mystery
Always can be solved no doubt.

You say you smoke Fatimas, they’re your type
But Sunday night I came and found a pipe.
Crossword Mama, you puzzle me,
But Papa's gonna figure you out.