Solution
o First try to set w/ screw driver, if not screw, set w/ nail set and hammer. Once set, check above and below to make sure you didn’t expose any more tightening the drywall to the stud.
o 2nd fill the hole w/ spackling (for smaller patches) Let dry then sand.
o W/ a small nail hole patch, there should be no need for texturing, however, if it is smooth wall texture, I would recommend a second coat of spackling, that covers the hole, smoothed out to hide any bump, but beyond the edges of the patch.
o Prime the patched area keeping it smooth and feathering the edges, then paint. If you skip the primer, the spackling will absorb the paint and after three or four coats, it will still be discolored.