Cut off separating or bubbling drywall tape with a sharp utility knife. Scrape off old, crumbling mud with a drywall blade, then cut the tape back to where it is still firmly attached to the drywall.
Turn on a heater and close the windows to give the wallboard behind the tape time to dry if it is wet.
Scrape over the damaged area again with the drywall blade to remove any last chunks of old mud, then spread a coat of new mud with the blade. Spread it liberally, working it into holes or cracks.
Moisten a length of paper drywall tape and lay it over the mud, then scrape the blade over the tape to flatten it and squeeze out excess mud from behind it. Hold one end of the tape to hold it in place while you scrape in one direction, then scrape over the end you were holding. Let the mud dry overnight.
Spread a second coat of mud, using the blade to flatten the edges of the mud into the wall. Let this coat dry overnight, then repeat with a third coat. At this point, the repair should be blended into the rest of the wall, but if it isn't, spread a fourth coat after the third has dried.
Sand the repair lightly, then paint it, and any discolored areas surrounding it, with drywall primer and a paintbrush.