Then-senior constable Kristian James Samuel White fired his service Taser at Clare Nowland after being called to the Yallambee Lodge aged-care home in the early hours of May 17, 2023.
The 35-year-old was handed a two-year good behaviour bond and ordered to complete community service in March after a jury found him guilty of manslaughter.
Justice Ian Harrison decided the need to deter other police officers from committing similar offences played "only a minor role" in his decision.
But prosecutors believe he made a mistake in not issuing a strong warning to other police officers.
They have lodged an appeal against White's "manifestly inadequate" sentence on four grounds, including the judge's assessment of the importance of general deterrence.
Prosecutors also claim Justice Harrison erred when he found that White made a "terrible mistake" but his crime was less serious than other manslaughter cases.
The judge was wrong to conclude a jail sentence would be "disproportionate" to the seriousness of the offending, prosecutors allege.
They are also challenging Justice Harrison's conclusion that prosecutors agreed with White's lawyers that the police officer honestly believed firing the Taser at Mrs Nowland was necessary to negate the threat he thought she posed.
Prosecutors argued during White's trial that the use of force was "so obviously excessive" that no reasonable police officer would have made the same decision.
Mrs Nowland, who suffered from undiagnosed dementia and weighed less than 48kg, was holding a knife and using a walking frame when she encountered White.
White drew his weapon and pointed it at her for a minute before saying "nah, bugger it" and firing the weapon at her chest. The great-grandmother fell and sustained a fatal brain injury.
Her eldest son Michael said White's sentence was "very disappointing".
"A slap on the wrist for someone who's killed our mother - it's very, very hard to process that," he said.
White wrote a letter to the Nowland family in which he took full responsibility for his actions.
"I deeply regret my actions and the severe consequences it has caused (sic)," he wrote.
"I have not had a single day go by where I haven't thought about the incident that morning and what could have been done differently."
White has been sacked from the NSW Police force but requested the decision be reviewed.
The appeal against White's sentence will be mentioned in court on April 17.
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