Welcome to Blender! Be warned, cutting holes or indents into curved surfaces is a surprisingly difficult problem in hard-surface modelling. You can take a look at a YouTube search for "blender cutting holes in curved surfaces" to get a sense of how complicated it can become.
You can move your vertices onto a perfect sphere using the Mesh -> Transform -> To Sphere tool. However, applied directly using the usual "median as the pivot point", it would tend to deform the whole mesh toward your indentation (because there are more vertices there). To apply it without modifying the rest of your mesh (much), and assuming the UV sphere's origin is still at the center of the sphere (should be, if you haven't moved it since adding the UV sphere mesh), you might want to try the following steps:
- Move the 3D cursor to the UV sphere origin, if it's not already there. To do this, in Object Mode, with the sphere selected, use ShiftS -> Cursor to Active.
- Change the pivot point to the 3D cursor. To do this, use "period", and select 3D Cursor.
- Apply the "To Sphere" transform to all the vertices. To do this, switch to Edit Mode, make sure you're in vertex selection mode with 1, use A to select the whole mesh, and use ShiftAltS for "To Sphere", and drag the mouse right until the "To Sphere" amount in the top left of the window hits 1.000, then left-click.
- Change the pivot point back to median: To do this, "period", then select Median Point.
That should place the points firmly on the sphere. It may slightly increase or decrease the radius of the rest of the UV sphere, but it shouldn't deform it.
However, this probably won't fix your rendering problem as well as you're hoping. There will still be pinching and minor imperfections even with all vertexes lying precisely on the sphere. To get rid of these or at least minimize their effects, you'll need to learn about good mesh topology and the roles of quadrilaterals versus triangles and n-gons (faces with more than 4 sides) as well as "poles" (vertexes with more or less than 4 associated edges).
But, for a quick and dirty project with a couple of minor rendering blemishes, "To Sphere" might be good enough. Good luck!