Burton is accused of attempting to murder a fellow inmate by stabbing him at Auckland's maximum-security prison at Paremoremo on December 20.
He is serving a life sentence with a minimum non-parole period of 26 years for killing Lower Hutt man Karl Kuchenbecker in January 2007. He was on parole at the time after serving 14 years for murder after stabbing Paul Anderson to death outside a Wellington nightclub in 1992.
The Dominion Post reports today that the documentary series, Beyond the Darklands, analyses notorious criminals and their crimes.
The episode about Burton, who had a leg amputated after being shot by police when arrested during the Kuchenbecker incident, is due to be screened on TV1 on Wednesday.
Burton's lawyer Peter Tomlinson said he had applied to the court for the injunction, and hoped the case would be heard today.
Depositions for the attempted murder charges are set down for June.
