Bishop Stephen Cottrell, who will be confirmed as the 98th Archbishop of York next month, was approached about the safeguarding issue 10 years ago but did not pass it on to the authorities.
The Church of England said an investigation concluded that Bishop Cottrell poses no current risk of not responding appropriately to safeguarding disclosures.
Bishop Cottrell said in a statement: “Ten years ago, I was approached about a safeguarding allegation regarding a priest.
“I was able to see the survivor and begin to hear what was a difficult and harrowing story.
“However, I was moving between roles at the time and, although I did speak with colleagues about the actions that needed to be taken, I failed to ensure that these were properly documented and followed through in the way I would expect.
“Now that I have discovered that this incident was not followed up as it should have been, I am deeply distressed and extremely sorry.”
He continued: “In my new position as Archbishop of York, it is absolutely essential that I am open and transparent about the need for the whole of our Church to be scrupulously honest with each other about any failings in safeguarding.”
Bishop Cottrell said good safeguarding is an “absolute priority” for the Church of England, which needs to stop being “too quick to protect its own reputation and slow to admit its failings”.
He added that he now has a greater understanding of why safeguarding is important after working with survivors in the Diocese of Chelmsford over the last decade.
He said: “Therefore, although I am embarrassed that I did not follow this up as scrupulously as I should have done 10 years ago, I want to go on the record about what has happened in order to demonstrate a new spirit of openness and transparency over how we ensure that the Church is as safe as it can be, that survivors are listened to and dealt with honestly and perpetrators brought to justice.”
The National Safeguarding Team (NST) said concerns were raised earlier this year after information came to light from a clergy file.
The NST said: “The concerns raised were about the action taken following allegations of domestic abuse perpetrated by a parish priest.
“At the time, Bishop Stephen responded to the survivor, offered support and subsequently referred the allegation within the diocese, but did not ensure the matter was referred to the statutory authorities or directly to the diocesan safeguarding adviser.”
It said the bishop had shown “insight and humility” in accepting his failings, which were due to a lack of training and understanding.
It added: “The NST investigation concluded that he posed no current risk of not responding appropriately to safeguarding disclosures and that informal action was a reasonable and proportionate response to the case.”
The Archbishop of Canterbury, Justin Welby, said he is looking forward to working with Bishop Cottrell in his new role and prayed that this experience would “strengthen his commitment to safeguarding and ministry”.
He said: “He clearly should have informed the authorities and made fuller notes of what he did in this case.
“He has shown humility in immediately admitting he failed to act as he should have done.”
He added: “I am also reassured that he did refer it on and saw the significance of offering support and contacting the survivor, who must always be the priority.”
Bishop Cottrell is due to be confirmed as the Archbishop of York in a virtual service on July 9.