The head of the national inquiry into child abuse has been urged to publicly address claims that it covered up an alleged sexual assault by its most senior lawyer.

An inquiry worker alleged that Ben Emmerson QC sexually assaulted her in a lift at the inquiry’s offices in Millbank in central London in early September, according to BBC Newsnight.

He was suspended that month, over what the inquiry said at the time were leadership concerns, but the suspension was lifted the next day when he resigned, allowing him to keep working for the inquiry for two months.

Emmerson’s lawyers have said the allegation of sexual assaultis “completely false” but the claim has added to the turbulence afflicting the inquiry.

Tim Loughton, a Conservative MP who was chair of the home affairs select committee when it questioned the inquiry chair, Prof Alexis Jay, and other panel members last week, said Jay needed to address the allegations and restore confidence in its work.

“It is slightly ironic that an inquiry into cover-ups about historic child sexual abuse is handling things in this way and [it] does need to be much more transparent to give assurances to those many thousands of people who were affected that this is being looked at properly and it’s fit for purpose,” Loughton told BBC Radio 4’s Today programme.

“She [Jay] really does need to put on the public record the assurances that this is being dealt with properly and that the inquiry is doing its important work properly to the satisfaction of the survivors who are the most important people in this.”

The committee asked Jay a direct question about Emmerson’s suspension last week but the allegations aired by Newsnight were not mentioned by Jay. When it was put to him that they should have been, Loughton stressed the independence of the inquiry from political interference and said there may have been legal issues at work. But he insisted that the inquiry must “step up its game”.

On 28 September, Emmerson was suspended and the next day he quit. Referring to Jay’s plans for a review of the approach of the inquiry, he said: “I had my personal doubts about whether I was genuinely the right person to steer that review process. Since then, it has become clear to me that I am not the person to take this review forward on your behalf.”

At the time Jay paid tribute to his “enormous contribution to the inquiry”.

His suspension was lifted and, at last week’s meeting, Jay told the committee that Emmerson was continuing to work for the inquiry from home, writing a handover document for his successor, who is yet to be announced.

The Labour MP Lisa Nandy told Newsnight that the claims surrounding Emmerson and a possible cover-up were “the latest and most serious to date in a series of allegations that have emerged into the public domain in recent weeks”.

Including Emmerson, three senior lawyers have quit the inquiry since Jay took over as chair from Dame Lowell Goddard, who resigned in August.

Earlier this month, it was alleged that Goddard was racist and abusive to staff on the inquiry. She has denounced the claims as “completely false”.

A lawyer for Emmerson said: “Mr Emmerson categorically denies any allegation of sexual assault, or bullying or other misconduct at the inquiry.”