"He told police he had shaken Lincoln intentionally ... he knew you're not supposed to shake babies because it could kill them."
He also admitted to hitting Lincoln on the head about three or four weeks before his death, she said.
Kos said Wakefield began dating Lincoln's mother while she was pregnant with him. Throughout the pregnancy he was "touchy" about the fact the baby was not his biological child.
After the baby was born in January 2018, the mother had less to do with her family, and they saw little of Lincoln.
He could at times be "quite controlling" of the mother and "cruel" to her and the child.
Wakefield convinced the mother to return to her work as a hairdresser, while he stayed at home and looked after the child.
On June 11 he called her at work saying Lincoln was not breathing.
Fire and Emergency staff arrived at their Upper Hutt home and provided care to Lincoln, who was hypothermic, and rushed him to hospital where he died the next day.
Kos told the jury a person does not have to intend to kill someone to be guilty of murder.
If they intentionally cause bodily injury to a victim while knowing it could cause death, they can be found guilty.
Defence lawyer Steve Gill said Wakefield had already pleaded guilty to the "totality" of his offending by admitting the manslaughter and wounding charges.
"The Crown cannot possibly prove the necessary intent in this case," he said.
"It's all about what's going on in the defendant's head at the time of the killing, not later when he [gave the police interview]."
The trial is set down for two weeks.
