Is there a specific term for the practice of prefacing a statement with another statement that is in contrast (in spirit) to what is being said?

There is a related concept called apophasis, where you deny calling something to attention, but nevertheless bringing up the subject. But the intent is different, since the purpose here is to offer a pre-emptive denial before bringing up something negative or controversial.

**Examples**:

I am not a racist, but this neighborhood would be better with less [x] people.

We shouldn't gossip, but did you hear what Joseph said to Mary?

**Related concept**: Is there a related term for prefacing a criticism with an unrelated positive statement so as to reduce the negative impact of the criticism?

**Example**:

A corporate email from an office manager that begins with "I hope everyone is having a great Monday." before lambasting the employees for not cleaning up after the Friday office party.