The word “grace” suggests some kind of reference to scripture, as in the following from the New Testament:
Bear with each other and forgive whatever grievances you may have against one another. Forgive as the Lord forgave you. (Colossians 3:13)
When suggesting ways in which to display grace to a speaker who experiences a mishap, it is important to first consider the nature of the mishap. Did the speaker misquote someone? Did the speaker slip and fall? What exactly is meant by “mistake?” A member of an audience might view as mistaken an expression of an opinion with which that audience member disagrees. That would not necessarily be a mistake on the part of the speaker and could actually indicate an undue level of arrogance on the part of the observer or listener. If one assumes the more benign definition of the word “mistake,” the matter is rather simple. If the mistake involved an objectively assessed misstatement, like if the speaker misstates an easily verifiable fact or misquotes another individual, an expression of grace would involve either ignoring the error or, following conclusion of the talk, approaching the speaker and asking to speak to them for a moment. At that time, and with some discretion, the audience member could say, “Excuse me, but I couldn’t help but wonder if the exact quote was x," or, in another situation, “Excuse me, but I think the actual information is x.” This would not appear confrontational (assuming the proper demeanor on the part of the audience member) and might represent the start of a meaningful dialogue with the speaker.
If the option of approaching the speaker is not available for any number of reasons, then requesting contact information for the speaker from the individual who arranged the session is an option. With that contact information, the audience member could send a letter or email to the speaker expression reservations about something the speaker said or did. It would then be upon the recipient of the communication to show grace by responding politely.
Some venues offer opportunities for audience members to ask questions of speakers. This is a very public situation and one that demands civility. At that time, the audience member could politely state that they disagreed with something the speaker said and offer an alternative set of facts or opinions. The speaker could then respond if they desired or simply acknowledge the audience member’s point of view and move on to the next questioner.
Displaying grace or proper manners is not a challenge. Despite the extraordinarily exaggerated rhetoric surrounding most public speakers these days, there is no reason not to respond to mistakes with grace and style. Corrections should not be demeaning or condescending; they simply and politely reflect one’s desire to correct a mistake based on one’s own observations and experiences.
Now, consider the possibility that "mishaps" literally refers to a physical act, such as falling or sneezing at an inopportune time. Under such circumstances, and assuming that one is close enough to the speaker to assist, approaching with an offer to help would certainly be graceful, as would an expression of sympathy or understanding. Again, it all depends on the mishap. A sneeze during a talk would require nothing more than saying, "bless you." A fall might require medical attention or at least assistance standing up. Either way, the circumstances should dictate the response.