On the first day of his trial at the Old Bailey, Renshaw, of Skelmersdale, Lancashire, admitted planning to kill Labour MP Rosie Cooper with a sword-like Gladius knife last summer.
The 23-year-old defendant also pleaded guilty to making a threat to kill police officer Victoria Henderson.
Judge Mr Justice Jay directed the jury to deliver a formal guilty verdict on the first two charges facing Renshaw who is also on trial alongside Christopher Lythgoe, 32, from Warrington.
Lythgoe denies giving Renshaw permission to murder the West Lancashire MP on behalf of National Action on 1 July last year.
Renshaw and Lythgoe, along with Garron Helm, 24, of Seaforth, Merseyside, Matthew Hankinson, 24, of Newton-le-Willows, Merseyside, Andrew Clarke, 33, and Michal Trubini, 35, both of Warrington, also deny membership of the banned far right group.
The then Home Secretary Amber Rudd banned the group in 2016, which meant that being a member of or inviting support for the organisation is a criminal offence and carries a maximum sentence of 10 years in prison.
Prosecutor Duncan Atkinson QC, opening the trial, said: “This case concerns these defendants and their support for, involvement in and membership of the proscribed racist neo-Nazi group National Action.”
Renshaw planned to carry out a “politically and racially motivated murder” in support of National Action, he told the court.
Jurors were told they were done “with the blessing” of his leader, Lythgoe.
The court was told the group had embarked on a “campaign of virulent anti-Semitic and homophobic propaganda” since 2013″.
Mr Atkinson said the group tried to recruit and radicalise young people through violent images and “hate-filled language”.
The group was banned when it supported the murder of MP Jo Cox in June 2016, Mr Atkinson said.
He said despite the ban, the defendants remained active members of the organisation after it was proscribed.
Evidence would come from a “disenchanted” former member of National Action, Robbie Mullen, who passed information to Hope Not Hate, Mr Atkinson told the court.
The defendants were part of the North West area branch of National Action which met at the Friar Penketh pub in Warrington, jurors heard.
Lythgoe pledged to continue National Action after the ban and said in an e-mail last December: “Long term we’ll keep moving forward just as we have been.
“We are just shedding one skin for another.”
Mr Atkinson said Renshaw’s threat of violence was personal as well as political.
He was arrested in January last year on suspicion of stirring up racial hatred in two anti-Semitic speeches in Blackpool and at the Yorkshire Forum for Nationalists.
Detective Constable Victoria Henderson and another officer interviewed him before he was released on bail.
After examining his phone, police revealed what they considered to be evidence of child sex offences and he was interviewed again by DC Henderson, jurors heard.
Renshaw revealed his murderous plan to National Action members at a meeting in the Friar Penketh, on 1 July, the court heard.
It is alleged he told members he had already bought a machete marketed as offering “19 inches of unprecedented piercing and slashing power at a bargain price”.
Renshaw had also researched “cutting the jugular artery” on the internet last May, the court heard.
Mr Atkinson said: “Renshaw’s plan had a more sophisticated dimension in that its objective was not simply to make a political point – as he put it to kill for National Action and White Jihad – but to revenge himself on those he considered to be persecuting him and trying to send him to prison for a significant period.
“Renshaw explained that, after killing Rosie Cooper MP, he would take some people hostage and would then demand of the police when they attended that DC Henderson come to the scene.
“His plan then would be to kill that officer who was, he said, his real target.”
Lythgoe’s alleged response to the plan was to tell Renshaw: “Don’t f*** it up.”