how to set and align the c-plane to the center of selected points?

There is no problem to set the c-plane to the center of the polygon. But often you need to set the c-plane to the center of a hole or completely unrelated points.

Is it possible to do this?

What I found - 1) I can set the c-plane to the center of the selected points, but the orientation will be completely arbitrary. 2) I can turn on the advanced mode and set the c-plane to the correct plane by three points, but then the c-plane will jump to the middle point and will not be in the center. That is, either the correct position, or the correct orientation.
Does anyone know a way to set the c-plane to the center of the selected points and set the orientation to the average normal of these points?

Unfortunately, you might just have to create extra geometry as a side branch, and use the middle position as your second selection of your orientation picking.

Not sure if this is what you are looking for. But procedurally solved for the points to give avg normal and position and copy a grid to this averaged point. Used the grid to align and orient the c-plane.

I think the quick-and-dirty solution here is:

Polyfill that hole - single primitive.
Use 'Quick mode' for the Construction plane orientation.
Enter orientation mode, click on the new poly to orient.
Delete polyfill node.

But I think a more general solution would be tricky as I think you would have to calculate a plane of best fit through a set of arbitrary points - they may not be planar, and the 'centre' of those points could be defined in multiple ways.