How does one trust one’s children again after a blunder like that?
First, you have to determine whether it was a deliberate and conscious lie. If that was the case, then I would tell them that I had lost trust in them and cannot accept their word for anything, that they would have to work to regain the trust.
If it was a question of perception, then you have to apply the "Wisdom of Solomon" to extract the truth from their accounts (taking into account other independent data sources).
So, for example, if my child tells me something bad about another child, or a teacher, or coach, I always FIRST, before I do anything else, or react in any way, get one or two independent accounts of what ACTUALLY transpired. [THESE lessons have been learned, often the hard way, after the first few children]. I invariably find that their story doesn't line up perfectly with reality, since there's a certain amount of subjectivity, and the degree of subjectivity (imposition of interpretation and imagination) is very high among children. So I'll get another kid's account, and if possible also an adult's account, preferably from someone without any kind of personal "stake" in the situation. You can usually get very close to the actual truth by comparing the various accounts ... and filtering out bias. So, for instance, I have a couple of my kids who are always plotting to get the other one in trouble, so I know that I will get accounts that are biased against their enemy as much as possible.
I think that it's important for us parents not to go in after our children's teachers, priests, etc. "guns-a-blazing" after having accepted one child's account at face value. It used to be that if a child got disciplined at school, the parent would double up on it at home. Now, when a school disciplines a child, the parents get up in arms and call for the heads of the teachers. If my child gets in trouble, my first question is, "So what did you do?" It's important for that first reaction NOT to be "How dare that teacher do such and such!" That seems to be the knee-jerk reaction these days.
And, you know, so what if a teacher is not a perfect human being? I fail often as a parent, and these teachers are helping to raise my children. I am grateful for their efforts, and tolerant of their failings, and realize that they are sinful human beings just like the rest of us. It's rather hypocritical that parents who themselves fail all the time feel that they have a right to demand perfection of their children's teachers, and the first reaction is to call for their heads while brandishing torches and pitchforks in response to even the slightest mistake or slip-up. These teachers are undertaking their vacation to help educate the children often at great cost to themselves, and we owe them more than this.