I seem to recall (something) (doing something)

The phrase "I seem to recall..." means that you think that you remember something. You can use this phrase if you're not sure about what you remember:

I seem to recall him telling me about some video game that he was into, but I don't remember which one.

You can also use "I seem to recall..." when you are sure about what you remember, but it seems like other people have forgotten it:

I seem to recall you telling me that you were going to send me those pictures. What happened to that?

(someone) promises to (do something)

Use this phrase to describe a promise that someone made:

Didn't she promise to help you?

You can also use "promised not to ___":

What the hell! You promised not to tell him.

cut the deficit

When a government spends more money than it collects, that is called a "budget deficit" or just a "deficit".

The phrase "cut the deficit" means to reduce the difference. For example, if the government was spending $100 billion per year more than it was collecting, and now it's only spending $50 billion more per year, then a politician might say:

We've cut the deficit in half.

Or:

We've cut the deficit by 50 percent.

come through (on something)

If someone makes a promise, and they successfully keep their promise even though it was difficult or in doubt, then you can say that that person "came through":

She said that she'd make it to my show, and this time she came through.

Aside from a promise, you can also "come through" on someone's wish or hope. For example, you might talk about a sports team this way:

Man, that was a really tough match, and I didn't know if they'd be able to pull it off, but yeah, they came through!