The teacup handle doesn’t look like it’s attached to the cup. Is it a separate mesh?
There are a couple of tools that might help with those circle inside a square teacups you’ve got there.
First, if you select some vertices and press SKey then X or Y or Z then Zero, the vertices will line up perpendicular to whichever axis you picked. Useful for getting flat sides without disturbing the rounded corners.
Then, if you select a loop of vertices (such as the squared off outer edge, duplicated and scaled in a bit) you can turn it into a circle using the “to sphere” button on the mesh tools panel. Put the cursor in the center of the circle, and use the cursor for the scaling pivot.