One of the children is now dead and the other recovering in hospital in Melbourne after their mother allegedly tried to kill them that afternoon in the River Murray at Moama in NSW.

The 27-year-old mother has been charged with the murder of her five-year-old son and the attempted murder by drowning of her nine-year-old son.

The body of the five-year-old boy was found in the Murray River, near Moama, on Saturday morning.

The mother refused to appear via video link in the Deniliquin Local Court on Monday morning where, in a brief hearing, the charges were revised to formally include murder.

Her lawyer, Peter Hebbard, asked the court if a psychological assessment of his client could be completed before the next court appearance on May 2.

Outside the court, he told reporters that his client had been crying and sounded "very remorseful", but still didn't understand what had happened.

"The words she used to me was 'I would never hurt my children, I do not know what happened'," Mr Hebbard said.

"She knows that something's happened and she knows she's done something. She was crying and she wanted to know how the little fellow was going down in Melbourne.

"I told her to the best of my knowledge he was okay, that her mother was down there with him and she seemed happy with that."

Lawyer, Dale Brooks, who represents the boys' grandmother, said his client was "absolutely devastated".

"There was ample time to save these children," Mr Brooks said.

"There's a running joke here among criminals that if you want to avoid police you simply head for the border."

The two boys had lived with their grandparents in Deniliquin for more than 18 months, and were only reunited with their mother in the past few weeks.

The grandparents are understood to have been forced to take out an apprehended violence order against their daughter last May and had recently alerted NSW Police, the Department of Family and Community Services in NSW and Corrective Services about her deteriorating mental state.

Mr Brooks said several "warning signs" had been ignored.

"[The mother] was obviously a very troubled woman, the mother of these kids, and there just does not appear to have been enough action taken about some of these early warning signs."

NSW Police has been contacted for comment.

NSW Minister for Family and Community Services Pru Goward said her department was undertaking a comprehensive review as a result of the incident.

"The death of this little boy and the serious injury suffered by his older brother are a tragedy," Ms Goward said. "This incident has shocked us all, and [Family and Community Services] staff are supporting the family.

NSW homicide detectives are expected to complete their brief of evidence by April 17.

The woman handed herself into police in Echuca at about 10pm on Friday, and is understood to have made a range of admissions before she was extradited to the NSW town of Deniliquin.

The woman is understood to have collected her two boys on Thursday evening and taken them fishing at a spot on the Murray River near Moama.

It has been alleged in court that she first attempted to drown the older boy, who managed to break free when his mother was distracted by her younger child's terrified screams.

As he returned to the riverbank, the older brother was mauled by a dog. He was later flown to Melbourne's Royal Children's Hospital, where he remains in a stable condition.

Bernadette Murphy, principal of St Michael's Primary School in Deniliquin, where the younger boy had only recently joined classes, said the school community was devastated.

"It is a dark time for students, staff and parents," Ms Murphy said.

"While he had only been in our school community for six weeks, the start of his learning journey has left a footprint in all our hearts.

"We are a close-knit community and this tragedy has had a profound affect that will require much support."

Mr Brooks said he would hold Premier Gladys Berejiklian to account on her pledge to investigate the government agencies responsible for dealing with the boy's mother before the tragedy.

"A lot of these departments express their condolences and then go to ground and deny any responsibility for what happened. It's just not good enough," Mr Brooks said.

He also called for greater co-ordination between Victorian and NSW police on the border.

At the weekend, the father of the two boys, who lives in Brisbane, launched a crowd funding campaign to raise $5000 to fund a trip to Melbourne to visit his injured son.

He has not seen the boy for about six years and never met his five-year-old son who drowned in the Murray River.