The leader of the House of Commons, Andrea Leadsom, has launched an investigation into allegations of bullying of Commons staff by MPs, a move that will put increased pressure on the Speaker, John Bercow.

Leadsom announced the investigation after an urgent question from MPs prompted by a Newsnight investigation alleging that Bercow’s private secretary resigned and had post-traumatic stress disorder after working in his office.

The allegations against Bercow surfaced shortly before Debbie Abrahams, the shadow work and pensions secretary, was suspended from her role while an allegation of bullying against her was investigated. Abrahams hit back at the claims on Monday night, counter-accusing members of Jeremy Corbyn’s office of bullying her.

Corbyn’s office was said to be concerned about Abrahams’ alleged treatment of a number of both current and former staff members. She is believed to have been notified that there were formal allegations against her only in the past week, and was not given details of the specific accusations or who had made them.

Some shadow cabinet ministers have suggested that the counter-claims of bullying centre on Corbyn’s chief of staff, Karie Murphy, which was strongly denied by the leader’s office.

Debbie Abrahams. Photograph: David Gadd/Sportsphoto Ltd/Allstar

Abrahams will be subject to Labour’s internal code of conduct, which can result in individuals requiring employment training, a national executive committee reprimand or not being allowed to use Labour party staff. No formal complaint was submitted by Abrahams following her statement.

Some MPs have given their backing to Abrahams, including the former shadow equalities and women’s minister, Kate Green, who tweeted: “Debbie is my friend. But were she not, it still can’t be right that she’s not apparently even been told details of complaint against her.”

On Monday, Bercow chaired the urgent question on bullying, tabled by the Green party co-leader Caroline Lucas, who had requested that a deputy speaker take his place.

“Reports of bullying of house staff are of huge concern to me and to honourable members right across the house,” Leadsom said. “I am committed to stamping out all kinds of bullying and harassment in parliament.”

Leadsom said she had hoped to cover parliament staff under an independent grievance procedure set up by a cross-party working group at the end of last year, but had been advised that staff were already covered under the House of Commons’ respect policy.

“Following the Newsnight allegations and others, it is clear the respect policy may not be sufficient to protect house staff,” Leadsom said. She said she would propose a short independent inquiry to the House of Commons commission, the body that runs the Commons administratively, when it meets on Monday.

The commission is chaired by Bercow, but when Leadsom was queried about his presence on the commission by the Conservative MP Michael Fabricant, she said the investigation would be led independently. “It is right that everyone working for and with parliament, regardless of position or seniority, should have the same rights and protections and should be held to the same high standards,” she said. “Make no mistake, there is a need for change.”

The Conservative MP James Duddridge said he was shocked that Bercow was allowed to chair the debate. “Is it appropriate for Mr Speaker to remain in his place while there are allegations against him which he is trying to suppress using taxpayer-funded money through sending out letters through speaker’s counsel?” he asked Leadsom, who reiterated the need to take complaints seriously without directly responding to the query.

In the Commons, Lucas said it was clear staff members should be covered by the new independent procedure. “There are now considerable grounds to assert the respect policy does not have the confidence of the staff it is intended to protect,” she said.

Several Labour MPs spoke out in Bercow’s defence during the debate, including the shadow House of Commons leader, Valerie Vaz, who questioned whether any formal complaints had been received. “It is noted you have granted the urgent question even though you are one of the people mentioned in media reports,” Vaz said. “No one, critics or otherwise, can deny your commitment to accountability and transparency and attempts to move the house forward.”

The Labour MP Paul Farrelly, another of those accused of bullying Commons staff in the Newsnight report last week, urged fellow MPs not to “jump to judgment”. He called the BBC report “one-sided” and said the investigation should approach the cases in a “more balanced way”.

Bercow has strongly denied the claims that he bullied his former private secretary Kate Emms, who worked for him between 2010 and 2011.

Bridgen, who is pro-leave, admitted that his motivation to move against Bercow was partly down to his views on Brexit. “Given his apolitical position as Speaker of the House of Commons, he shouldn’t be able to trumpet them from the rooftops as he does.”