Jurors in the Lindsay Clancy murder trial on Wednesday afternoon again told the judge that they were unable to reach a unanimous decision. The judge then delivered a set of instructions, sometimes called a “dynamite charge,” in a last-ditch effort to push them toward a verdict and avoid a mistrial.

It was the second time in two days that the jurors alerted the court that they were struggling to agree on whether Ms. Clancy should be convicted of murdering her three children.

The jury of nine women and three men, who have now been deliberating for nearly 30 hours, were brought into the courtroom just after 2 p.m., uniformly looking grim and stone faced. They appeared to listen to these new instructions with rapt attention.

“You should consider that it is desirable that the case be decided,” Judge William Sullivan said, “and that you have been selected in the same manner and from the same source from which any future jury would be.”

Earlier on Wednesday, a woman accused of intimidation in connection with the trial entered a not guilty plea a day after her arrest outside the courthouse. Her arraignment came moments after the judge in the Clancy case questioned jurors and then sent them back to continue deliberations.

A lawyer for the woman, Dawn Light, 56, said she had been trying to get an image of Ms. Clancy on her phone and did not mean to film jurors. Judge Sullivan questioned jurors briefly under oath before sending them back to the deliberation room.

Ms. Light, who is from Sutton, Mass., was arrested outside the courthouse on Tuesday and charged with “intimidation of a witness, jurors or person furnishing information in connection with criminal proceedings,” according to a statement from Sgt. Gregory Jones, a spokesman for the Massachusetts State Police.

The prosecutor in Ms. Light’s case said during the arraignment that she had been videotaping jurors in a restricted area, in violation of Judge Sullivan’s orders. “The sanctity of the judicial system is at risk” because of Ms. Light’s actions, said the prosecutor, Nicole Piacentini.

Ms. Light’s lawyer, Jennifer White, said she was a retired nurse with no criminal record. The judge overseeing her arraignment, Shelby Smith, denied the prosecution’s request for $50,000 bail but ordered Ms. Light to stay away from the courthouse unless she was appearing in her own case.

In the Clancy courtroom, lawyers from both sides witnessed the judge’s interaction with the jurors, whom he questioned one by one, but public spectators were unable to hear what transpired, and it was unclear exactly what the questioning concerned.

Several jurors shook their heads and clearly answered “no” in response to the judge, who on Tuesday had encouraged them to keep trying to break their impasse and dismissed them at the end of the day with the words, “You’ve come this far.”

Failure to reach a verdict would result in a hung jury, with neither a conviction nor an acquittal. The judge could then declare a mistrial. Prosecutors would decide whether to try Ms. Clancy all over again, in front of a different jury, for first-degree murder or less serious charges. Considering their options could take up to a week, or more.

Kevin Reddington, Ms. Clancy’s lawyer, said on Tuesday that he doesn’t think Ms. Clancy is emotionally prepared to endure another trial but that he would continue to represent her.

When a reporter asked on Tuesday if Ms. Clancy would consider pleading to a less serious charge, like manslaughter, Mr. Reddington said, “Cruz drives the bus,” referring to the county’s district attorney, Timothy Cruz, who brought the original case.

Ms. Clancy, 36, doesn’t deny killing her children with exercise bands in 2023 at their home in Duxbury, Mass., an affluent suburb of Boston. But she has pleaded not guilty, and her lawyers have argued that she should not be held criminally responsible because she was suffering from postpartum psychosis.

Mr. Reddington has said that she tried repeatedly to get help when she began to experience intrusive thoughts after the birth of her third child. But she was overmedicated, he said, and hallucinated a male voice commanding her to kill her children and herself.

Jennifer Sprague, a prosecutor, told jurors that Ms. Clancy had made a series of premeditated choices before killing the children, Cora, 5, Dawson, 3, and Callan, 8 months. Those included asking her husband to leave the house and deciding where and how to kill the children, as well as whom to strangle first.

Ms. Clancy’s husband at the time, Patrick Clancy, said he had found her lying injured in the backyard after the killings. Ms. Clancy had cut her wrists and neck and jumped from a second-story window of their home, leaving her partially paralyzed.

The trial has drawn crowds of demonstrators, including many women who see Ms. Clancy’s struggles to get help as an indictment of the health care system’s approach to maternal mental health care.

For a guilty verdict, the prosecution must prove beyond a reasonable doubt that Ms. Clancy both committed the crime and was criminally responsible for it.

Judge Sullivan has instructed jurors that Ms. Clancy would not be criminally responsible if she had a mental disease or defect that prevented her from controlling her actions or from understanding that they were legally or morally wrong.

If jurors find Ms. Clancy guilty, they must choose among first-degree murder, which would send her to prison for life, or less serious charges of second-degree murder and manslaughter, which could carry significant prison time but also leave open the possibility of parole.

If found not guilty by reason of insanity, Ms. Clancy would be sent to a secure psychiatric institution.

Jacey Fortin, Jenna Russell and Remy Tumin contributed reporting.