The phrase with prejudice is a legal term used when a judge makes a ruling on a case and declares that it is final. A lawsuit dismissed with prejudice cannot be filed again.
With prejudice sounds like it means "with bias." But the words are used a bit differently in the context of the law. In many situations, when a judge dismisses a case because of issues with the process, the people involved are allowed to try again by refiling. That's called dismissal without prejudice. But when the judge says, "No, I'm done with this, and you can't bring it back to me or any other judge later," that's dismissal with prejudice.