The abuse of children and vulnerable adults has been "rampant" in British institutions, and the "church's failure to face the misconduct of those in its service is inexcusable," the Archbishop of Canterbury has said.

In a private letter to the mother of three boys who were sexually abused by the headmaster of a Church of England school, Archbishop Welby apologised for their traumatic experiences.

"I read your story with the same sense of sadness and dismay that I have felt on far too many other, similar accounts. The betrayal of Christ in such behaviour is complete; the church's failure to face the misdeeds of those in its service is inexcusable," he wrote to Marilyn Hawes.

"I can only apologise for what happened then, and for what has happened now, most sincerely and with deep sorrow."

Welby added that abuse has been rife "in a huge number of institutions and localities...[but] that is not in any way mitigation or excuse for the church."

"It is also clear that there is a very significant legacy of unacknowledged cases in the Church of England," he said, noting that "all necessary steps" are being taken to address the issue.

Welby has repeatedly apologised to victims of sexual abuse at the hands of Anglican clergy, and pushed for a public inquiry into the church's failure to prevent it and respond effectively.

Last year, he branded the issue "absolutely agonising", and said a "complete change of culture and behaviour" in the Church is needed.

"We cannot in 20 years be finding ourselves having this same debate and saying 'Well we didn't quite understand then'," he said during a meeting of the General Synod in York in June.

"There has to be a complete change of culture and behaviour and in addition there is a profound theological point. We are not doing all this, we are not seeking to say how devastatingly, appallingly, atrociously sorry we are for the great failures there have been for our own sakes, for our own flourishing, for the protection of the Church.

"We are doing it because we are called to live in the justice of God and we will each answer to Him in our failings in this. Accountability is one we must take with the utmost seriousness."

The Bishop of Manchester, Right Rev David Walker, issued a statement over the weekend praising victims of sexual abuse for their bravery in coming forward.

His statement came in response to a report into allegations against the now-deceased former Dean of Manchester, Robert Waddington, who sexually and emotionally abused young boys in his care.

"My heart goes out to those whose lives have been irreparably damaged by what Robert Waddington did to them. When I read of the ongoing effects of his abuse, decades after it took place, it makes my blood run cold," Bishop Walker said.

He thanked Archbishop of York John Sentamu for the inquiry into Waddington's behaviour and the Church's complicity, but also praised those who bravely gave evidence against him.

"I want to say thank you to those who Robert Waddington abused, and who came forward," he said.

"Thank you, because every time a survivor of abuse speaks out it makes it just a little easier for the next person to speak. Thank you for the courage you have found."