As Donald J. Trump’s Manhattan criminal trial enters a third and crucial week, new witnesses will shed light on a hush-money deal struck in the final days of his 2016 presidential campaign. The judge will also weigh a request to hold the former president in contempt.

Mr. Trump, the first former president to face criminal prosecution, is accused of falsifying records to cover up the hush-money payment, which was made to a porn star, Stormy Daniels. The $130,000 payment — made by Mr. Trump’s fixer, Michael D. Cohen — silenced Ms. Daniels’s story of a sexual encounter with Mr. Trump.

The judge presiding over the case, Juan M. Merchan, is expected to rule as soon as Tuesday on the prosecution’s request that he hold Mr. Trump in contempt for repeatedly violating a gag order barring the former president from attacking witnesses, prosecutors and jurors. The prosecutors requested a $1,000 fine for each of Mr. Trump’s 10 statements that they say ran afoul of the order — including attacks on Ms. Daniels and Mr. Cohen, as well as the jury.

Also on Tuesday, prosecutors are expected to wrap up their questioning of Gary Farro, a banker who helped Mr. Cohen open the account that he used to pay Ms. Daniels. Mr. Trump’s lawyers will then cross-examine Mr. Farro.

Once Mr. Farro completes his testimony, the prosecutors might delve deeper into how the hush-money deal was made. One potential witness who was involved in the deal is Keith Davidson, the lawyer who represented Ms. Daniels and negotiated the payout with Mr. Cohen.

Other potential witnesses include Mr. Cohen, Ms. Daniels and Hope Hicks, a spokeswoman for Mr. Trump during the 2016 campaign and in the White House.