The letter states: “The RUC could have taken steps to provide dedicated patrolling in the area when a local resident reported suspicious activity and a breach of the peace line.
“That dedicated action may have prevented Paul’s murder.”
It goes on to state the investigation “did not identify or pursue all key lines of inquiry”, including a failure to arrest three potential suspects.
The PSNI also apologised for “significant disclosure delays” at the inquest.
The letter concludes: “The Police Service apologises unreservedly for the failings which impacted adversely both on the delivery of justice for Paul and on his family’s quest for the truth.
“We are committed to improving how we help families and communities impacted by the Troubles by being more open, transparent and victim focused and by disclosing as much information as we possibly can.
“The Police Service believes that further information about Paul’s murder can safely be released to Eugene and hopes that the pending Supreme Court judgment will allow for this to happen.”
The Committee on the Administration of Justice (CAJ) supports the family’s case.
Its director, Daniel Holder, said: “Eugene and before him his and Paul’s mother Margaret, have long fought for justice and truth.
“It’s welcome that the chief constable has quickly delivered this apology in relation to the RUC and the PSNI’s failings.”
He added the family is entitled to the release of further information and noted the outcome of June’s Supreme Court hearing is still awaited.