A grandmother admits she killed her 2-year-old grandson, who died from severe head injuries, but denies she is guilty of murder, a court has heard.

Kathleen Elizabeth Cooper, 65, is on trial before a jury and Justice Sarah Katz in the High Court at Auckland.

Cooper is alleged to have murdered her grandson Jermain Mason Ngawhau at her Clendon Park flat.

Her lawyer, Paul Dacre QC, said his client was responsible for killing an innocent child, but had shown no murderous intent and was therefore not guilty of murder.

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Crown prosecutor Aaron Perkins QC told the jury that Cooper, who earlier pleaded not guilty, assaulted Jermain on December 13, 2015.

He said when emergency services arrived paramedics determined the toddler needed immediate surgery.

"He had a severe head injury, emergency surgery was performed at Starship [Hospital], but there was no hope - the injuries were too severe."

Jermain died on December 18, 2015, when his life support was switched off.

"The damage had been done," Perkins said.

He added doctors found the toddler's brain was severely swollen, with bleeding on or around the surface of the brain.

"The entire left side of his brain was effectively dead already.

"A very bleak picture, indeed."

Perkins said an expert medical witness will suggest the injuries were more commonly associated with high-speed vehicle accidents.

Cooper had been caring for her daughter's four children at the time of Jermain's death, all of whom were pre-school age.

"At some stage in 2014 these children were effectively taken [from their mother] ... [Cooper] effectively put her hand up to look after and raise those four children," Perkins told the court.

He said witnesses will speak of the way Cooper would discipline the children, with allegations from neighbours of smacking, constant yelling and slamming doors.

He said there will be evidence that Cooper appeared kind-hearted towards the children.

"What was seen externally by people wasn't necessarily a reflection of what was going on behind closed doors at home.

"It was something of a mixed bag."

He said staff at the day care centre which helped look after the children will testify that they noted "quite regular and sometimes quite significant bruising on the children".

Perkins said it was Cooper's "overall attitude" towards Jermain which was important.

"You will hear that Jermain had some significant developmental issues.

"He was also very small for his age," referring to the toddler's weight of just 10.3kg and height of 79cm.

"Very light and very short ... At the time of his death he was still having a great deal of trouble walking.

"The Crown suggests that Cathleen Cooper was struggling to come to terms with Jermain's difficulties," Perkins said, adding she showed a "lack of understanding or sympathy for his position".

He said Cooper had said Jermain's difficulties were "down to the fact that he was lazy".

"The defendant would be very angry if there were any toileting accidents."

Perkins said Cooper also showed signs she was smoking methamphetamine at her home.

"Police organised for hair samples to be taken not only from Jermain but from the other three children," he said, adding scientists found that the hair showed traces of meth.

"This may well have contributed to the frame of mind that [Cooper] was in.

"Her inability to make allowances for his developmental problems and to cope with those."

Perkins said Cooper insisted to police that she had phoned the ambulance first after discovering Jermain's injuries.

"But she didn't, members of the jury, that is very clear from the telephone records."

Cooper also blamed Jermain's death on her then 4-year-old granddaughter, whom she claimed hit the 2-year-old over the head with a computer tablet, Perkins said.

"[She] tells many people, basically anyone who would listen."

Perkins said police, as part of their investigation, also started listening to Cooper's phone calls in early 2016. The phone calls will be played before the court.

"I know that f****** little b*** did it," Perkins said Cooper uttered in one conversation, referring to her granddaughter.

However, Perkins said the 4-year-old girl told police that "nana threw [Jermain] in the hallway, because he was being naughty, he was not walking".

"She pick him up and throw him in the hallway [sic]."

Perkins added Cooper also claimed Jermain's injuries may have occurred when "someone else has come inside [the house] while she was asleep and assaulted Jermain".

The prosecutor said there would be several exhibits produced throughout the trial showing the extent of the injuries the 2-year-old suffered.

Cooper showed no murderous intent - defence

In his opening statement, defence counsel Paul Dacre QC said the death of a young child was "obviously a tragedy".

"It's a tragedy that a young boy hasn't fulfilled his potential, he's been taken away from his family and he's been taken away from the community."

However, he accepted that his client was responsible for killing an innocent child.

"She's responsible for the death of her grandson Jermain," he said.

"Mrs Cooper accepts that she caused the death and that the death was caused by an assault."

But, he said, while Cooper accepted she was guilty of the manslaughter of Jermain she had shown no murderous intent and was therefore not guilty of murder.

"You've got to look at the circumstances and look at what was going through her mind."

He said the burden was on the Crown to prove that Cooper knew her actions were likely to cause death.

The trial is expected to last about two weeks.