Dr Hollingworth receives a reported annual allowance of $360,000, plus travel, staff and an office in Melbourne.

The Greens' Rachel Siewert introduced legislation on Wednesday to allow the minister or parliament to cancel an allowance to a former governor-general or spouse when either had engaged in "serious misconduct".

She said she expected the legislation to capture the case of Dr Hollingworth, who has been the target of child support groups over his handling of abuse allegations when he was Anglican Archbishop of Brisbane.

The bill looks unlikely to get major party support after a similar attempt earlier this year by then senator Derryn Hinch was voted down.

Dr Hollingworth's two years as governor-general were clouded by criticism of the way he handled child sex abuse allegations, leading eventually to his resignation.

In 2017, the royal commission into the handling of child abuse was critical of Dr Hollingworth's actions, including his decision to allow an accused paedophile to remain as a priest after he admitted to abusing two boys. John Elliot was eventually convicted in 2002 and 2003 of child sex offences involving seven boys.

Dr Hollingworth was told about the allegations against Elliot in 1993, with one victim saying he had been abused nearly weekly over four years from the age of about nine. Elliot admitted the abuse, and a psychiatrist advised Dr Hollingworth that he was untreatable and there was a real risk that he would offend again. Dr Hollingworth did not report it to police and allowed Elliot to continue as a priest for another four and a half years until he retired, under conditions, and then allowed him to work as a locum following his retirement.

Dr Hollingworth told Elliot that sacking him "could cause unintended consequences that would make things worse for you and the church".

Dr Hollingworth said he believed the offending had been isolated and Elliot had promised not to have contact with young boys.

But the royal commission said Dr Hollingworth had failed to take into account the psychiatrist's advice, and made "a serious error of judgment" focusing on Elliot's needs instead of the victim and the protection of children more generally. The royal commission also criticised a letter Dr Hollingworth had sent to the victim's brother telling him that "the Christian rule is one of forgiveness and reconciliation".

"Dr Hollingworth's letter to [the victim's] brother was inappropriate and insensitive. These responses demonstrate the way in which the principle of forgiveness can be used to discourage a complainant, whether a survivor or a third party, from pursuing their complaint," the commission found.

In 2016, Dr Hollingworth wrote to the victim to apologise for his "poor handling" of the complaint.

"After a great deal of consideration over the past 22 years, I acknowledge, unconditionally, that my actions were misguided, wrong and a serious error of judgement that I genuinely regret," he wrote.

Senator Siewert said there were no provisions to stop allowances to governors-general when it was no longer in the public interest.

"A former governor-general or their spouse will continue to receive retirement allowances worth millions of dollars, even where they have engaged in serious misconduct including conviction and imprisonment for a crime," she said.

Her legislation defines serious misconduct as "inappropriate, improper, wrong or unlawful conduct" and gives examples of corruption, sexual misconduct, sexual harassment, theft, fraud and other criminal behaviour. It extends serious misconduct to "an omission to act", which Senator Siewert said would send a "clear signal to people in power that the cover up of abuse and illegal behaviour is never acceptable".

While Dr Hollingworth's handling of the case was criticised by the royal commission he has not been accused of a "cover-up" or any illegal behaviour, nor any of the examples set out in the bill.

A coalition of child support groups wrote to the prime minister last year asking for Mr Hollingworth's allowance to be cut off.