Is there a revolving small flat file for a drill?
It sounds like what you are describing here (and in the other thread) is called the "seat" for the valve stem washer, and usually that seat is replaceable for this exact reason. If it is replaceable, the inside will have a geometric shape, usually hexagonal, sometimes square (or the corners of a square cut into the sides of a round opening, see below), to accept a tool that us used to unscrew and remove it. Here is an example of a bad faucet seat, one with a "groove" in it like you describe.



You typically cannot see anything but the end when you remove the valve stem, so you may not have known that this is a separate piece. But if you look on this photo, the insides are hexagonal, so you can use a hex key tool to remove it. They are almost always made of brass so that they do not corrode and become impossible to remove, but it sometimes takes some effort. Once you do so, take it to a hardware store and you can buy new seats, they are cheap. They will come in pairs, replace them both because if one is corroded, the other one probably is not far from it too.

Here is an example of the type that needs a square tool.