At the insertion point there will be two BoS crates with supplies in them, so make sure to check them. Both objectives will be inside the depot, which has multiple entrances. There are two well guarded entrances southern corner (marked 4 and 3 on the map) which have both posted and patrolling guards. On the Northern side there is another entrance with much less resistance but is only accessible through a minefield. Then there is the tunnel entrance (marked 2 on the map), This is fairly well guarded by 4-5 super mutants but once accessed will take you to the heart of the depot. If you do take this path, be careful going up the ladder at the end as there may be super mutants standing nearby.
Once inside the base there will be two staircases (marked 6 and 7 on the map) to the lower level you need to access. Since the West staircase just loops around to the Eastern staircase, it is faster to use the Eastern one. If you do use the Western staircase, in the room full of propane tanks, shoot the tanks from as far away as possible to avoid being caught in the explosion. After the propane room or the first room from the Eastern staircase, you will will encounter Toccomatta, who will be squatting on the floor, non-hostile. When you talk to him, he will mistake you for Paladin Latham and will reveal that General Barnaky is not in Osceolla and has be captured by a new opposing force and foreshadows a later threat. Once he is finished talking he will collapse dead. On his corpse will be Barnaky's letter, a note from Barnaky to his wife, Maria. Now there will be two entrances, an opening to the left and a door at the top.
Do not go through the door at the top as it is a trap. In the doorway on the left there will be a super mutant with a flamer thrower waiting. So as soon as you enter the room, run to the bottom edge, causing him to be unable to reach you with his weapon. Once this room is clear, the next will be the final room with Latham and a super mutant waiting for you. They will both fire simultaneously so approaching the automatic door in the prone position is suggested. Once he is dead the mission will be complete, but before you leave, make sure to check the (easy to miss) chest at the foot of the bed. Inside will be a briefcase, which upon inspection by the Scribes will turn out to be a remote nuclear detonator.