Robinson, wearing a shirt and tie, sat next to his lawyers in a courtroom located six kilometres from where Kirk, 31, an ally of US President Donald Trump, was shot dead on September 10.
Robinson's father, mother and brother were also present, his lawyer Richard Novak told the court.
The proceedings in Provo, Utah, before District Court Judge Tony Graf went into closed session shortly after starting.
The public, including Robinson's relatives, was asked to leave temporarily, and a video feed was shut off.
During the closed session, Graf and the lawyers were to discuss what could be made public from previous closed sessions.
The judge, who is weighing a request from Robinson's lawyers to ban cameras from the courtroom, said he would rule on the access questions later on Thursday.
Robinson is accused of firing a single round from a rooftop that killed Kirk during a Turning Point USA event on the campus of Utah Valley University in Orem, about 65km south of Salt Lake City, as Kirk debated with students. Kirk's death sparked denunciations of political violence across the ideological spectrum.
Since Robinson's first court appearance via video link on October 27, Graf has made rulings designed to protect his presumption of innocence in a case he said had drawn "extraordinary" public attention.
The judge ruled Robinson could appear in court in regular clothes but must be physically restrained.
Graf prohibited media from filming or photographing Robinson's shackles after his lawyers said images of him restrained and in prison clothing could prejudice jurors.
One videographer and one photographer have been designated by the court to share images and audio from the courtroom with other news outlets.
Erika Kirk, Kirk's widow and now head of his conservative Turning Point organisation, called for cameras to be allowed in the courtroom to preserve transparency.
Robinson is charged with seven criminal counts, including aggravated murder, obstruction of justice for disposing of evidence, and witness tampering for asking his roommate to delete incriminating texts. Prosecutors said they would seek the death penalty.
