The Add/Delete Anchor Point tools won’t help; that’s strictly for vector paths. The center point of a layer is moveable if you get the cursor directly over it. You should see the small icon at the lower right of the cursor when you’re over it. Try this: select the Move tool (V) and in the option bar at the top of your window, turn on “Show Transform Controls.” That should give you a bounding box and center point. Move the cursor over the center point and the cursor should change from a pointer with a large 4-point arrow to a pointer with a small 4-point arrow to let you know you’re on the center point. You should now be able to drag it around.
If you still can’t affect the center point, go to the tool options (far left of the option bar) and click on the drop-down arrow. Then select the small gear icon and choose “Reset Tool” from the menu. That will reset the Move tool to its standard settings. Might be enough to jog loose whatever the problem is. If that still doesn’t work, you may need to reset Photoshop’s preferences. On Windows, that’s holding down Ctl-Alt-Shift while starting Photoshop. Might be you have a corrupt preference.
That said, Photoshop will NOT remember center point placement once you complete a transformation. If you move the center point to, say, a person’s eye, then rotate and hit Return, if you come back to that layer to transform it again, the center point will be back at the center of the layer, not where you left it.
—
It is easier to destroy than to create.
More fun, too.