We're talking about rebuilding trust here. The parent-child relationship is primary, and for whatever reason, your child has lost some trust in you. It's not a being angry about discipline thing, it's not something frivolous or superficial, it's not a lot of things. It is making some mistake -- that you may or may not have realized at the time -- that has deeply hurt your child.
Hurt is irrational
Your child is hurt and you feel bad -- and those hurt feelings may come out in irrational ways. No one ever claimed children were rational, logical or even-tempered beings to begin with. Add in a serious hurt and there's potential for some serious outbursts. Even though the hurt in this case is very personal to you, do try to separate the outbursts from the core emotion and issue. It can be easy to get defensive in the face of such irrationality, so it will take great care not to. And don't lash back.
Careful about guilt
It may seem incongruous, but you need to be careful about acting out of guilt. Yes, you feel guilty, but giving gifts or allowing activity because of that guilt may only send a message to your child that he or she can manipulate your feelings. Think about what you would give or allow if this relationship didn't have this issue and act accordingly. You can't buy trust.
Keep talking and listening
Keep talking to and at your child. This doesn't mean apologizing over and over. It does mean continuing the effort to communicate both about resolving the issue, rebuilding the trust and about everything in your regular day-to-day life.
Keep listening, too. And listen between the words. Initially your child's words may be all anger and hurt, but there could be more subtext than initially obvious. What else could be going on? Is there something more that you can address that can help this repair along?
In the aftermath of the hurt, get back to the consistency of your life with your child as much as possible. Keep to the normal routines of your family life and parenting style. Demonstrate by your actions that you are going to continue to parent to the best of your ability.
Rebuilding trust with your child will take time. It will take persistent, consistent effort on your part to reestablish the baseline of the relationship. If your child is older -- a teenager, perhaps -- it could take even more time. Do not give up. More than ever, treat that child like you would want to be treated, like you would want your parents to do for you, and do not give up.
Mistakes in parenting do happen. While we hope they are few and far between, each parent does at times have to rebuild trust with a child. Whether your child is school age or a teenager, focusing on being the consistent, thoughtful parent you are at your core can help both of you move beyond the hurt to a stronger parent-child bond.
More on the parent-child relationship
When your teenager hates you
5 Tips for stress-less parenting
5 Things a step parent should never do