With a specific example, I’d have a more certain answer.
What it reminds me of is when someone blurts out a really powerful thing in a non-committed way in a non-committed scene, and it’s too sad/mean/real to just ignore.
I’m going to risk trying to make up hypothetical situations and then honestly say what I think I would do, rather than come up with some proper-sounding teacher platitude.
1) Let’s say a nervous actor is a priest, and even though the scene was mostly about how this priest is jealous he’s not allowed in the choir, he then blurts out “Well, I’m just going to diddle an altar boy anyway” and the audience goes “ooooh.”
Same if a person accidentally makes a sexual innuendo, the audience notices it and then the person acts like they meant to make that innuendo. Like if that priest character said “When I need guidance I look to the children” and the crowd laughs, and only then the actor says “to molest them.”
I would either:
a) React big, abandon the previous scene and focus entirely on that confession for the rest of the scene, and try to have it work through surprisingly intense commitment or
b) Treat that confession like a sub game. Like I keep talking about the priest wanting to be in the choir and then now and then throw in “and of course we’ll have to deal with your molesting at some point, also.”
c) If the actor REALLY just threw that away and it was clear he wasn’t committed, I might just straight-up dismiss it. “You wouldn’t have the balls, Father Klein. Now let’s get back to this choir thing.” That’s lame but I might do it.
I wouldn’t ignore it, and I wouldn’t do politically correct scolding either. First is a cop out, second is no fun.
2) The case of over emphasis, like the very common case of a doctor’s scene where the doctor is telling you that you are going to die in five minutes, or that you have AIDS or cancer, and it’s so huge that it feels like there’s nothing to do but scream.
a) React big and scream and have the size of my reaction be the focus of the rest of the scene.
b) React but react smaller than you really would, so that you can still have a conversation with the doctor. “Oh my God, ten minutes? I can’t believe it!” Just choose to not be a screaming mess for the sake of having a scene.
3) The real best strategy is to be an amazingly good actor and make the situation true in your brain and also instantly realize how you would really feel and say that. I’m saying this in an overstated way, but there are performers who have such gravity in their voices and can process their feelings so quickly, that they would take either of the above cases and respond with such truth and specificity that it would turn what feels like a gaffe and make it an amazing scene. They do it in one line. They turn to the priest and say “We have to stop such things” and the room is thrilled.
P.S. I have one slightly bitchy thing to say, and I recognize that you’re just sending a question to an improv blog and not parsing every word you’re using but I think this is an important point so I’m going to make it anyway: be wary of the part of your brain that says “get roped into” — though I sympathize, it’s not helpful. Your job as an improviser is to welcome being in situations that you weren’t expecting. The audience enjoys it when you’re thrown, you should enjoy it too. If you field it right, you got roped into the most interesting scene in the show.