When building your brand you will try things that will not work and you should learn from your mistakes and press forward. When moving forward you have to make sure that your brand did not take a major hit because of the mistake. Protecting your brand is as important as building it. When you are trying to become an expert in your field you have to be strategic in all things and that includes damage control.
There are a few things that you can do to address the mistake professionally:
Own It - As soon as the mistake is brought to your attention, own it. People respect you more if you are open from the start and just admit it. There is nothing worse than someone trying to cover up the mistake and denying it only to be caught lying later on. Trust is golden in your personal brand. If the audience can not trust you then how can they support you. Build your brand in integrity not deception.
Apologize Publicly and Privately - Once you have owned the mistake, make a statement of apology. Go to the website, blog, organization, etc. and place the statement there. Judging on the depth of the mistake a video may be appropriate. Only use the video approach if it looks sincere, if it looks rehearsed or staged then it could cause more harm than good. The statement shows that (1) You are aware of the situation (2) You are taking responsibility and (3) You are have the integrity to apologize publicly. Then you should send a private and personal apology to the people that were affected by the mistake. Do not expect an answer back just know that your intentions were good and you did the right thing.
Correct the mistake if possible - Sometimes it could be a mistake like typos in an article or posting the wrong video and these things can be corrected and should be done immediately. Sometimes it may be a statement said or a video captured and these can not be corrected but they can be used as learning experiences and article topics so that others can learn from it as well. The best way to learn from a mistake is to turn it into a positive experience for someone else.
Mistakes happen but if you own the mistake, apologize publicly and privately then correct the mistake if possible then you have shown your professionalism and character. Many people do not take the time to do the things to preserve their personal brands. Be the stand out brand and display the personal touch.