Yvette Cooper has clashed with the chair of the national child abuse inquiry, warning her that transparency is crucial to maintain public confidence after two years of problems within the investigation.

Cooper, the new chair of the powerful home affairs select committee, has rejected a demand from Prof Alexis Jay – the inquiry’s fourth chair – that MPs desist from calling her or other inquiry members to give evidence before them. In a letter to Cooper published on Wednesday, Jay said it was important for the inquiry to maintain its independence and to be seen to be doing so.

“It is for this reason that I would urge the committee to consider carefully before requesting that anyone from the inquiry attends to give further evidence,” she said.

But Cooper made clear that transparency was vital, and that the committee would call whoever it wished to give evidence as the £100m public inquiry proceeded with its work.

“Given the two-year history of problems in the inquiry, it is vitally important that there is some transparency over the things that have gone wrong in the past and the way it is working now, so people can have confidence that problems are now being resolved and the work is back on track. It will be for the committee to decide when and whom to call on to give further evidence,” Cooper said.

Alexis Jay. Photograph: Martin Hunter

Jay has completed a review of the inquiry’s work and, in an update this week, revealed that further public investigations may not take place. The inquiry has set up 13 investigations that are intended to lead to public hearings with witnesses giving evidence on oath. These include investigations into the Roman Catholic church and Church of England, Lambeth council, Medomsley detention centre, Nottinghamshire council, Westminster and the late Lord Janner.

When the inquiry began Goddard promised up to 25 public hearings would take place, but Jay said the inquiry would commission new investigations “only if we consider they are necessary to fulfil our terms of reference”.

Investigations which may not now take place include into the BBC and Jimmy Savile, as well as into the Ministry of Defence and the armed forces.

On Wednesday Cooper repeated calls for the outgoing chair Justice Lowell Goddard to give oral evidence before her inquiry, something the judge has repeatedly failed to do. Goddard, who resigned in the summer, in written evidence to Cooper published on Wednesday claimed she had been driven out by a media campaign against her, saying no issues of leadership were raised with her by the secretary of the inquiry during her time as chair.

“A real and increasing strain particularly for me … was the intensifying media criticism of the inquiry which commenced around March this year. This centred on allegations of unfairness in the investigation of allegations concerning the late Lord Janner … and developed into widening personal attacks on me and my competence,” she said.

Goddard has denied claims made in the Times that she was racist and abusive to staff.

She said in her letter that the damaging impact of the continued media allegations led to the inquiry team losing its nerve about her ability to continue leading the public investigation into institutional abuse in England and Wales.

Goddard said she had agreed with Keith Vaz, the former chair of the home affairs committee, to produce written updates on the inquiry, but Cooper said that had never been the committee’s understanding.

She repeated a call for Goddard to give oral evidence, either in person, or via videolink to the committee. “We believe that an oral hearing would greatly assist our understanding of this complex and sensitive issue by allowing us to put precise and specific questions to you.”