The rattled minister told the media on Saturday afternoon she had received hundreds of messages of support, from the local health boards, clinicians and members of the public imploring her not to resign.

But she had not spoken to Premier Mike Baird since the latest scandal broke.

"I know the opposition has been calling for my resignation - that's pretty well expected - but I can tell you people in the health system, doctors, nurses, just some residents I met yesterday, said, 'For goodness sake, don't go anywhere'," Mrs Skinner said.

Fairfax Media revealed on Saturday that a second stillborn baby was cremated in error at Royal North Shore Hospital in 2012. At the time, the parents were assured that procedures would be improved to prevent a recurrence of the mistake.

Last week, NSW Health staff told a budget estimates hearing at State Parliament that a stillborn child had been mistakenly cremated after a mix-up by Royal North Shore Hospital workers in 2015.

Mrs Skinner said she had not learned of the second cremation error until she read about it on Saturday.

Mrs Skinner's office was aware of the incident as early as Thursday, when a media adviser, in a conversation with a Fairfax Media journalist, referred to questions that had been put to the Northern Sydney Local Health District about the matter.

However, Mrs Skinner's senior media adviser Liz Hannan said on Saturday she had not raised it with the minister.

NSW Opposition leader Luke Foley said a series of health system errors pointed to a "culture of cover up" in the minister's office.

They would never have been disclosed had it not been for doctors and patients coming forward, and the revelations that emerged from budget estimates hearings.

"The public can have no confidence that they're going to be told the truth about our health system under Mrs Skinner," Mr Foley said.

"This is about restoring confidence in our health system. That's why Mr Baird has to act this weekend to appoint a new health minister."

NSW Opposition Health spokesman Walt Secord​ told the ABC that Mrs Skinner's position was untenable.

"Another day, another error, another crisis in the health system," he said.

"The health minister said that human error happened. Yes, that's true. But sadly it is how you respond to these mistakes and how you assist the grieving families."

But Mrs Skinner said the system was more transparent than it had ever been, that 10,000 extra clinicians had been appointed in the last five years, and that the clinical excellence commission had been given extra funding to identify and prevent errors.

She also denied charges that she lacked compassion.

"Everybody knows that I have a real soft spot for families," Mrs Skinner said.

"Just about everything I do is based on what's in the best interests of patients and sometimes the best interests of patients not not revealing everything through the media."

A spokeswoman for Premier Mike Baird said: "He has full confidence in her and that hasn't changed."

Mrs Skinner told ABC Radio on Saturday that there would always be errors in the health system.

"These mistakes have been happening since time immemorial," she said.

"What we're trying to do is minimise these mistakes.

"Every incident is very regrettable, and that's why we're putting in place measures to try and make sure we learn from those mistakes and that we don't repeat them.

"These systems, unfortunately, always have these sorts of errors. The rate has remained pretty stable over the past two decades.

"What we're trying to do is investigate ways we can improve the system."

Macquarie University health systems analyst Jeffrey Braithwaite defended Mrs Skinner's decision to transfer responsibility for hospital management to Local Health Districts.

"No health minister in reality, no matter how good, can run things better than well-trained, motivated and professionally qualified local people can," he said.

"This is a $19.3 billion system with 108,000 staff treating 1.8 million patients. It can't be run out of the minister's office by a dozen or so minders supporting the minister, or a few hundred public servants in Miller Street, North Sydney.

"They are respectively responsible for health politics and health policy. Managing the system is the responsibility of the LHDs. If you do the knee-jerk thing and centralise responsibility every time you have a problem, you'll get what you wish for; a disempowered system with political or policy people remote from the patient and the clinical front lines making the decisions—which inevitably will be worse than the local decision making."

Revelations about the second accidental cremation come after the the death of one baby and serious injury to another at Bankstown-Lidcombe Hospital when they were accidentally given nitrous oxide instead of oxygen, an oncologist's flat dosing of chemotherapy at St Vincent's Hospital, another flat dosing doctor at St George Sutherland Hospital and over prescribing of antibiotics at RNS.