An example of such behavior would be the following.
The mother is depressed and alone with her child. She doesn't want to wake up the baby, but him being awake would really help her feel better. So she coughs a few times to wake him up. "Accidently on purpose".
You hate your workplace and you want to leave it, but you have feed your family, so you cannot just quit. Then you complain online using your real name on (for example) Reddit. Indeed, chances for your boss seeing it are realistically low. By the time he sees it, you've almost convinced yourself that it was not your intention. You "accidently on purpose" fired yourself.
And another one would be where you "accidently on purpose" hurt other people's feelings by mentioning things which can be understood in multiple ways. You can always say that "that wasn't your intention" or "it was just a coincidence I said that in front of you".
If there's no everyday word for this behavior, is there a psychological term for people who expose this trait?