I accidentally posted to Twitter what was supposed to be a direct message. Now, people are replying to me and asking me about that tweet.Can I delete it?
Deleting a status tweet accidentally published on Twitter
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You can delete your old tweets, just click on the trash can icon. However they may remain around for a while in searches and the like. They'll also stay on people's Twitter home page until they refresh the page.Voila! Gone forever... almost. Deleted updates sometimes hang out in Twitter search. They will clear with time.Source