Otherwise if you don't hide the sphere, you can uncheck the "Align to objects" option in the orientation picker when setting the cplane origin to avoid the snapping there, and then when drawing the arc instead of clicking on the point with the mouse (which will then try to align the arc to the plane of the sphere's seam edge which has its center point at that same spot) instead type in 0 and push enter to specify the construction plane's origin as the center of the arc, using numeric entry for that pick will not do any "temporary cplane" type snapping. Otherwise the newly drawn arc will basically try to stick in the same plane as the arc edge.

When you set the cplane it also affects how numeric input is used - when you type in coordinates they will be in that cplane's coordinate system. This is so you can temporarily use some local area of an object as if it were the world origin as far as entering in coordinates.

- Michael