No pic hard to imagine.
Gloss or emulsion?

Yes skim coat will help on the "potholes"
For the raised surface parts largely I feel its due to power point hacking run, and seems like raised surface. Better to level off those raised parts with rough (80-100 grit) sandpaper then skim coat "joint compound" or putty over.

Yes after skim just paint a foot or more like cloud bubble in comics
dont paint square box, very obvious. feather the edges.

I think the painter didn't have time to touch up after the hacking. Or the sequencing of events is that paint done first then only hacking was done..like XYZ come n dig road trenches after original tar.