This one hurts sometimes, especially when egos are involved, but if you screwed up, you really have to get it together and deliver a genuine apology. Forget the blame-transferring apology, too ("I'm sorry if you expected something different"). You want your client to know that you really understand what happened, why it went wrong, and where it started and that they can trust that it wasn't just a random blunder that you tried to cover up but couldn't.
If it was a failure on the part of multiple people, acknowledge that there is a process in place that needs to be fixed. Whatever the issue is, make them aware that you see it fully and are aware of the need for change.