Heart of the Swarm Campaign- Korhal: Planetfall (Part 3)

Sweeping North gives you an easy time taking out an Augustgrad gate, as well as the small Terran base positioned between, which is defended by soldiers, a pair of bunkers, and a single battlecruiser.  Wipe this out to cut further into the Terran attacks against your fortifications and you can sweep onward to the East from here.  Make sure you split your attention enough to build up the new base in the West side of the map as well, since enemy attacks are likely to keep coming for it.  Indeed, a force of foot troops, primarily marauders and marines, will soon be dropped just North of the fifth bile launcher, and head up the ramp to take it out immediately.  Fortunately, the gates are nearby, so it's just a moment's diversion to rescue the location and then return to your sweep across the map.

The last Augustgrad gate is partly protected by a large Terran base on an overlook to its East- you can take advantage of this yourself, though, by breaking into the base (which is remarkably lightly defended compared to the other locations on the map) and destroying it, then sweeping down the hill and simultaneously striking at the gate with ranged forces still in the raised area.

Once you've cleared out this location and defended the bile launcher last landed, Arcturus will call on another mercenary deployment, this one aimed to strike at Sarah herself.  Composed primarily of air troops, it should be pretty simple to take out the enemy forces and move on.  Take out the last holdout Terran troops in the area Northeast of that first potential mining outpost and you'll have cleared out the location for the sixth bile launcher- and paved the way for the Zerg assault on Augustgrad's fortress itself.

If you like, and if you have the troops remaining, you can tromp across the retractable bridge to the Northeast corner, where the last of Arcturus' forces are building up.  If you can take out the rest of the Dominion units here, you will have basically stopped most of Mengsk's assaults against you for the rest of the mission.  He will occasionally send small attack forces- mostly flying units, particularly vikings- against the bile launchers specifically, but can no longer mount regular attacks against your bases and is pretty much stuck with a loss here.