A single housewife?

I Agree you an American might say homemaker if she was single with a family but just her? unemployed.

It’s definitely unusual, but not impossible. Perhaps she is divorced, and the ex-husband is still paying her expenses? Perhaps she’s still raising the children from the failed marriage? It’s also possible she’s just unemployed and doesn’t want to say that out loud. Hard to tell without more details.

It wouldn’t be a typical thing to say but I can’t say its 100% wrong. Housewife is an old word where the -wife ending could mean “woman” as well as “female spouse.” Compare to words such as alewife, fishwife, spearwife. So it can be a woman who stays in the house. The word “woman” itself is from “wif” plus “man.” So anyway unusual but not crazy.

what is a "single woman's report"? it was offering employment as "housewife"? like that's the name of the job? I'm so confused about the context of this situation.

I would find that a bit strange myself. "Stay-at-home parent" would make more sense.