If you were to ask me what is the one grave mistake a great number of social media/digital managers make, I'd say it has to be posting a tweet that is meant for your personal account onto your client's account by mistake. I won't say all social media managers have done this, but many have and many will. Take, for instance, this post from the University of Maine's Parking Services department:
As far as Posting To The Wrong Account can go, this is among some of the worst I've seen, particularly because of the racial implications. Because it is irrelevant to what @UMParking normally tweets about, it's safe to say that this was likely a case of someone meaning to tweet from their personal account...and unfortunately, they posted a blatantly racist tweet onto their work account. Yikes.
Now most instances of Posting To the Wrong Account are not as terrible as the one I discuss here--normally it is as small as a colorfully worded tweet about your favorite football team or an "OMG" tweet about Scandal. But my point here is that it's a very common, very embarrassing mistake that can easily be avoided.
So what can you do? My one piece of advice is, unless it can't be avoided, never link your personal social media accounts, especially Twitter, to the same app as your client's account. In fact, I wouldn't even l wouldn't even log on to Twitter from your company phone unless you absolutely have to.
Trust me on this. It will save you a lot of emabarassment in the long run.
What do you do if you slip up? Delete the tweet IMMEDIATELY. Not 30 minutes later, not tomorrow, IMMEDIATELY, as soon as you realize you have made the mistake.
Then, depending on how offensive it is--and given the nature of Twitter, you will know right away--apologize to your Twitter followers. And not a "sorry if you were offended" apology. It can say something like this:
We have deleted the previous tweet - it was sent in error. We send our deepest apologies.
Now this won't necessarily stop all the outrage in your mentions, but you will have at least let your audience know that you are aware of the issue and that you are taking action. If other action is taken--i.e., a team member is removed from managing the account or is let go from the organization or company, you can note that as well.
But first and foremost, you have GOT to delete that tweet. There's no way around it.