Donald J. Trump faces a high-pressure day on Thursday as a crucial witness is expected to describe to the jury for the first time the hush-money payment at the center of the Manhattan criminal trial.

The witness, David Pecker, the former publisher of The National Enquirer, will take the stand for a third day on Thursday, and this time is likely to shed light on the circumstances surrounding the payment to the porn star Stormy Daniels during the 2016 election.

Mr. Pecker, whose magazine had previously bought and buried two other salacious stories on Mr. Trump’s behalf, decided not to pay Ms. Daniels for her account of a sexual encounter with Mr. Trump. Instead, Mr. Pecker is expected to explain how he and a top editor brought the story to Mr. Trump’s personal lawyer and fixer, Michael D. Cohen, who then paid Ms. Daniels $130,000 to keep quiet.

Mr. Trump, who later reimbursed Mr. Cohen, denies that he and Ms. Daniels had sex.

The Manhattan district attorney’s office, which brought the case, has said that Mr. Pecker was one member of a conspiracy that also involved Mr. Trump and Mr. Cohen. Mr. Pecker has supported that story, saying that the three men reached a secret agreement in 2015 in which The National Enquirer would promote positive stories about Mr. Trump and, importantly for the prosecution’s case, suppress negative ones.

Mr. Trump, the first American president to be criminally prosecuted, is charged with 34 felonies of falsifying business records related to the reimbursements to Mr. Cohen. If convicted, he could receive probation, or up to four years in prison.

It may be the only trial Mr. Trump faces before Election Day. Three other criminal cases are delayed, including one in Washington, where he is accused of plotting to overturn his 2020 election loss.