When dry, sand repaired areas lightly with 100 grit sandpaper or sanding sponge. Sand in a circular motion with light pressure so the area is smoothed but doesn't cause the mesh tape to become exposed. Pic 1.

If the mesh does show through, pat the sanded area with a lightly dampened sponge, reapply a thin layer of joint compound, let dry and sand again before proceeding.

If matching a ceiling with any other texture but popcorn, pat the sanded areas with a lightly dampened sponge and skim a moderately thick layer of joint compound across the entire repair, including the bare areas of the new piece. Feather patches onto the existing ceiling to conceal the lines of repair and let dry before painting. Pic 2.

For popcorn ceilings, pat repairs with a lightly dampened sponge after sanding, then apply the popcorn mixture with the 6" taping knife. The mixture is heavy and soggy, so apply a thin layer and let dry, then apply additional coats until the texture matches the rest of your ceiling. Pic 3.

No matter what finishing texture you use, let your repairs dry thoroughly, then prime and paint to match the rest of your ceiling.