I'm going to warn you about the click thing. You're developing a secondary behavior (or maybe it's already developed). I know when you're feeling the panic it's hard not to do what you instinctively do, but it doesn't help you communicate or stutter less. It's another layer of stuttering on top of the block. I'd just try to stick with it until the block resolves. I know some teachers, including college professors who stutter in front of the class. One I know taught for many years but was always a little self conscious about how his students felt. One year he decided to add another question to his end of the year evaluation form, Does my stutter affect bother you or affect your learning (he asked something to that effect). The overwhelming response was no, they didn't really care. It was only until that moment that he really accepted that people didn't care much and he finally stopped worrying when he stutterers. Now he just takes blocks as they come without stress (and not coincidentally his fluency has improved as well). I think it's good that you recognize this as a teachable moment. If you feel bringing it up could help ease the tension for you then that's a consideration. Your students might even be interested and ask questions.