AMNA NAWAZ: Welcome to the "NewsHour."

On the stand today for the fourth week of testimony in Donald Trump's hush money trial was the prosecution's star witness, Michael Cohen.

Trump's former attorney and fixer implicated Trump, testifying that Trump directed Cohen to pay adult film actress Stormy Daniels to bury her story on the eve of the 2016 election, and was then involved in the cover-up of those payments after he won.

William Brangham was in court today in New York, and he joins us now.

So, William, it's fair to say Mr. Cohen is arguably the most important witness for the Manhattan district attorney in this case.

This is the man allegedly at the center of those hush money payments and reimbursements.

What did he testify to today?

WILLIAM BRANGHAM: That's exactly right, Amna.

Prosecutors needed Michael Cohen to do basically two things today, first, as you were describing, establish that Donald Trump directed Michael Cohen to pay Stormy Daniels to be quiet so that she wouldn't spill her story and hurt Trump's chances to win the 2016 election, and then, second, to clearly implicate Trump in the scheme that prosecutors allege that they cooked up to falsify business records to then cover up the repayment of that money that Cohen paid to Stormy Daniels.

And on both those fronts, Cohen delivered somewhat dry, but pretty compelling testimony on both of those fronts.

AMNA NAWAZ: So let's take each of those in turn then.

Mr. Cohen has always said all along that Mr. Trump told him to pay Stormy Daniels to bury that story.

What else did he testify to today that added to that story?

WILLIAM BRANGHAM: That's right.

Cohen today really detailed the very elaborate attempts that Trump and his associates spent trying to keep stories of extramarital affairs from coming out during 2016 and the run-up to that election.

First off, he told more detail about the payments that Trump and Michael Cohen and David Pecker, the head of "The National Enquirer," paid to Karen McDougal, $150,000 to get her not to tell her story of having an alleged extramarital affair with the former president.

Trump was heard again on audiotape today saying that he knew how that payment was meant to go and that he would pay the money and how they would work out the checks.

We -- jurors heard that again from the president's mouth.

Then during the campaign came that infamous "Access Hollywood" tape.

And Cohen most clearly thus far has explained to the jury just how terrified the campaign was and the chaos that descended on them when that tape came out and how they tried to then brush it off as this -- quote -- "locker room talk."

Then, in the midst of that chaos, Stormy Daniels' story reemerges.

And Cohen described talking with Donald Trump at the time, asking him about this alleged Stormy Daniels story that was brewing in the background.

What did he know about it?

And Trump said this would be a disaster for his campaign if it got out, how it would hurt him severely with women voters.

He said -- Cohen said that Trump told him that: "Men will think it is cool, but women are going to hate me."

So we got a lot more detail on that front, how Trump directed Cohen to pay Stormy Daniels to bury that story.

AMNA NAWAZ: And then, of course, we all know what happens afterwards.

Mr. Trump goes on to win the election.

Michael Cohen then, according to this narrative, is still owed that $130,000 for the payment to Stormy Daniels.

And the core of the prosecution's case, William, is how Michael Cohen was repaid and how the business records were falsified to hide their true intent.

What did Michael Cohen testify to about that?

WILLIAM BRANGHAM: That's right.

This is the core of the case.

The 34 charges that Trump is facing of falsifying business records is how they repaid Michael Cohen that money and how they allegedly falsified those records.

And Cohen was worried and told in great detail about he was worried that he was never going to get repaid this money.

Trump had won the election.

Cohen was not being given a job in Washington, D.C., as he says he wanted.

And he was worried that he was going to get left holding the bag for this.

And he described this one meeting where he blew up at Trump's chief financial officer, Allen Weisselberg, and said, look, none of you guys stepped up to pay this money to protect the now-president.

I'm the one that did, and I need to be repaid.

And he and Weisselberg at that meeting then cooked up this scheme.

And jurors again today saw this quite striking document.

I want to pull it up for viewers to take a look.

It's a little bit confusing to understand what's going on here.

But this paper is a copy of the wire transfer that Michael Cohen sent to Daniel's lawyer.

And then you see all that handwriting at the bottom.

Again, there's a lot of figures and other terms in there.

The details are a little bit Byzantine here.

But in this document, on -- in Michael Cohen's handwriting and Allen Weisselberg's handwriting is exactly how they figured out in that meeting how Michael Cohen would get repaid.

And this document so far is the clearest piece of evidence yet of the whole alleged scheme that prosecutors say is going on here.

AMNA NAWAZ: So, William, if Mr. Cohen and Mr. Weisselberg came up with that scheme, does Cohen say that Donald Trump knew of or approved that scheme?

WILLIAM BRANGHAM: That's right.

Cohen says that, right after that meeting with Weisselberg, that he and Weisselberg went into Trump's office with that document.

And Cohen said it seemed to him like the two of them had already discussed this plan because Trump seemed familiar with the details.

But they went over all the details again, how Trump would -- how Cohen would get repaid in monthly payments, plus a bonus, plus some money he owed to a technical services firm, and, that in that process, Cohen says Trump clearly knew what was going on here and approved of the entire plan.

And so this is the essence of the argument, that the prosecutors allege that they falsified these documents to pretend this was all legal fees to Michael Cohen because they were trying to cover up that they were hush money payments.

AMNA NAWAZ: A big day of testimony today.

What about tomorrow, William?

What should we expect?

WILLIAM BRANGHAM: We're supposed to hear more from Michael Cohen.

There's a little bit more cross -- more direct examination of him by the prosecutors.

And then the cross-examination begins.

And this, as you and I have discussed, is where Michael Cohen will again get truly attacked, because, remember, he is a convicted felon.

He has several times pled guilty to lying under oath, to lying to Congress.

He's accused of election fraud.

He has changed his story about Trump's role in this alleged scheme multiple times.

He has also displayed a very clear animus to Donald Trump and has said he relishes the idea that Donald Trump will end up in prison.

All of that is likely to come out as the defense team tries to poke holes in this star witness.

We will see all that tomorrow, maybe Thursday as well.

AMNA NAWAZ: All right, that is William Brangham reporting live from New York tonight.

William, thank you.