Timing and place will be important. Only God, working through your good common sense, can impress you to know when and where that will be. But do confess your mistakes to your child. However, do not expect your child to readily respond in kindness and forgiveness. He or she will need time and some maturity to know how to respond.
Someone has to take the first step in reconciliation.
When you do your part, leave the response in the hands of God and in the will of your child. Don't dwell on the past. What has happened has happened. Jesus died for our sins. We need not nor cannot atone for them.