“Push deeper into the Hive realm to find the Deathsinger's ceremony.”
Once the bridge has formed, head across it. You can run and the cloak will just survive it. Stop at the end and let it recover (else you have a very high chance of dying).
On the far side you need to look around and see where the Hive are.
There are 3 Knights that patrol around the area on different routes. There is also an Orge that goes across the whole of the front portion. Watch out for this creature specifically. There are 2 Wizards flanking the door you need to get through. it is directly in front of you from the end of the bridge.
If, at any point, you need to retreat then you should head for the bridge. The bridge will remain active throughout the mission. None of the Hive will venture onto it so (unless you are detected). Take advantage of this fact.
The big concern here is getting past the Knight that patrols the entrance as well. You need to wait for the Knight to leave the doorway. Once it has cleared out, head straight for the door. You can just walk between the Wizards without being detected. If the Knight from the left is on the stairs then you can slip by it as well. It takes timing and paying attention to the detection radius but it can be done. Just hug the left wall and move as needed to stay undetected.
Once through the door, head down the left-hand corridor. Watch out for the pits. This is a safe place to run but remember to watch your cloak! If you de-cloak here there is a swarm of Thralls waiting for you. When you reach the dead end, you will want to drop into the pit in front of you.
Look around and find the exit. Remember this area for later. As you leaving, in the hallway, look to the left and note the big stone slab. Ghost can scan it and you can get some commentary from him.