Donald J. Trump’s criminal trial in Manhattan has reached the third week of testimony, and there are only a few key witnesses who have yet to take the stand.

There is Stormy Daniels, the porn star who accepted hush money in the waning days of the 2016 presidential campaign to keep silent about her story of having had sex with Mr. Trump a decade earlier. There is Michael D. Cohen, the Trump fixer who made that payment and is expected to be crucial for the prosecution. And there are several employees of the Trump family business, who helped the then-president reimburse Mr. Cohen.

Mr. Trump, the first American president to face criminal prosecution, is charged with 34 felonies related to those reimbursements: The Manhattan district attorney’s office says that he coordinated the falsification of documents related to the repayment.

The exact order of witnesses is unclear, but the trial is zipping along and the prosecution could wrap up in little more than two weeks. Ms. Daniels could testify as soon as this week, bringing her face to face with Mr. Trump, who for years has attacked her.

Mr. Trump has pleaded not guilty and has denied that he had sex with Ms. Daniels. If convicted, he could face probation or as long as four years in prison. He will not testify as part of the prosecution’s case; it is unclear whether he will later take the stand in his own defense.

Here’s what else to know:

• The false records charges: Prosecutors say Mr. Trump, 77, falsified the records to cover up his reimbursement of Mr. Cohen for the payment to Ms. Daniels, directing his company to describe the payments as “legal expenses” from a retainer agreement. The expenses and agreement were fictional, prosecutors say.

• Hope Hicks speaks: Mr. Trump’s former press secretary and White House communications director on Friday described the deep anxiety that gripped Mr. Trump’s campaign after the revelation of the so-called “Access Hollywood” tape in October 2016. In nearly three hours on the stand, Ms. Hicks also described concerns about a 2016 article in The Wall Street Journal, published days before the election, regarding Karen McDougal, a Playboy model who said she had an affair with Mr. Trump in 2006 and 2007. Ms. McDougal’s story was purchased by the parent company of The National Enquirer, only to be buried. Here are five takeaways from Friday’s court session.

• A gag-order ruling is pending: The judge, Juan M. Merchan, on Thursday held the trial’s second hearing on whether Mr. Trump had violated a gag order barring him from attacking witnesses, jurors, court staff and others. Days earlier, Justice Merchan had fined Mr. Trump $9,000 for nine violations and threatened him with jail if he continued to violate the order.