A mother accused of killing her two children left a note asking for forgiveness, a court has been told.
The mother was charged with the children's murder after they were found dead in the family home in NSW years ago.
The trio cannot be identified for legal reasons.
The mother appeared in NSW Supreme Court on Tuesday for a hearing over whether she should be found not criminally responsible for the deaths due to mental impairment.
The course of action was endorsed by both the woman's lawyer and the prosecutor, but Justice Richard Cavanagh questioned the strength of the evidence about her mental health impairment.
The mother's internet search history, a suicide note and seemingly contradictory psychological reports cast doubt on the extent of her impairment, he said.
"This is not a case involving schizophrenia, for example, there's no suggestion of deluded beliefs, there's no suggestion of psychotic episodes ... that so affected the defendant's thought processes that she wasn't able to reason", he said.
"It needs to be more than a mental health impairment."
The mother, who was found with stab wounds alongside her slain children, had penned a note that didn't seem to stack up with psychologists' findings, the judge said.
The note said she couldn't leave her children "alone in such a bad world without my support ... I hope you can forgive me".
Justice Cavanagh did not dispute the mother had major depression, but questioned whether that met the threshold for impairment.
"There's obviously a difference between her belief about her own attempted suicide and whether she believed what she was planning to do ... was wrong," he said.
He also highlighted the woman's internet searches prior to the alleged murder, which appeared to show she was "clearly planning the event".
But the woman's lawyer Madeleine Avenell SC argued her client's actions might not have reflected an understanding that killing her children was wrong.
"She knows that the action is wrong in the sense that she can say 'yes it's wrong to kill my child', but she didn't have the ability to exercise ... that wrongness," she said.
The hearing over the mother's mental health will resume on June 24.
Justice Cavanagh's concerns over the expert material were valid, defence solicitor Paul McGirr told reporters outside court.
"This is a very sensitive matter and it has to be done properly, particularly when you're dealing with extreme charges," he said.
The accused murderer, who dialled into court from custody, hung her head low during the proceedings, leaving only a tangle of hair visible.
