In the original story by Charles Dickens, Scrooge says that he promised Fan to take care of her son Fred. Scrooge makes reference to her dying after the birth of his nephew Fred. I don't think that he ever says when she asked for the promise to care for him or when he made it. It is left up to the conjecture of the reader.
That is easy to do in a book, but difficult to do in a movie because of the differences in the media. One can simply write a line of dialogue, such as: "She insisted that I promise to take care of Fred," in a written story and the author is done. The author has made the point that the character has a moral stake in caring for Fred, without any other specifications. For a script, the writer needs to leave it as a line of dialogue that is spoken, but because film is a visual medium there exists a strong temptation to create a visual scene. Once the screenwriter does that, there are multiple implications that fill in holes that the written story leaves open.
The best diplomat I know is a fully charged phaser bank.

Yes, I agree, it doesn't make sense. And when I read the original story, I found that the entire scene was not there. So the screenplay writers for this film basically made up the whole thing, and frankly did a poor job of it, and we're left to speculate why. I suppose their intent was to show what a cad Scrooge was, storming out of the room like that, before his sister even passed away. But again, that never happened in the original story, so her request for Scrooge to take care of her son, even though there is a father.oops. The screenplay writers were hoping you didn't catch that lol.

I believe she asked Ebeneezer to take care of her baby so that he would be involved in the boy's life and not blame the baby for her death, which he ultimately did. At this time it would not have been due to monetary reasons because Ebeneezer was still a clerk at Fezziwigs and hadn't yet sacrificed his heart to wealth.

Good point. Fan must have known that her hubby wouldn't be able to care for their child for a very long time.
Also, I agree with another possible explanation given in this thread - that Fan was hoping for
financial
support from Scrooge.

Jim Hutton (1934-79) & Ellery Queen
=

It doesn't really make sense. I concocted my own explanation: Fred's father must have died soon after Fan, so maybe he already had consumption and she knew he wouldn't live long, and Fred would soon be an orphan.
Flat, drab passion meanders across the screen!

I thought she meant to have the kind of relationship with him that he ultimately ended up having with Tiny Tim and the rest of the Cratchit kids. He wasn't taking Bob's place in raising them, but would see to it that they had everything they needed.

That's a great point jjk. Until you mentioned it, I never thought to consider the feelings of Fan's husband in terms of how Fan humiliates him. What a bad day for him. Scrooge snubs him and blames him for his wife's condition, then his wife disses him and dies immediately afterwards.

I always wondered about that scene too. Especially with the poor guy standing there and having to hear that.

Indeed, Scrooge was rich and Fan's husband was not.

I would guess that Fan's husband didn't have much money and Scrooge did.