"This is by no means all your fault, kid. Don't blame it on yourself. I made a couple bad decisions on the way as well. In fact, it's really the Caravanian's fault. This was all a trap you know." Raven spoke, trying to sound calm. His mind was racing as he said this, trying desperately to think of a way out of the situation. But for the first time, he had nothing. No wild plans, no tricks left up his sleeve, no weak points to punch. He was surprised at first when Ivan got so close to him, but it made sense as despite everything, Ivan WAS still just a scared child. He patted the top of the boy's head, in an attempt to be comforting. He clutched his amulet.
Then he realized something: The stream was probably the best bet for being safe from the mud demon, at least for a while. It couldn't be reached by foot however, and the branch did not go far enough. However... Would the amulet work on a material that wasn't a human being? Could he possibly throw it into the mud to create a patch of ground that the demon could not control? It had worked on cursed items to cleanse them of dark magic. It was worth a shot right? And even if it didn't work, maybe, just maybe, Raven could still get close enough to the water to throw Ivan into it.
"Ivan, I'm about to try something a little ridiculous... Okay, a lot ridiculous. But it logically could work and it means we have some small chance of living. We're not that high up, so jumping isn't going to hurt a man like me. If neither of us survive this... Well, I'll see you in the afterlife in that case. Just get ready to swim okay?" Raven whispered to Ivan. "Also, hold your breath."
Raven flung the anti-magic amulet in front of the tree, just nearby the stream. It landed in a mud puddle that was likely to be possessed. Raven ran to the end of the branch and leaped into said mud puddle (while holding Ivan) where the amulet landed and, with much of his leg length buried in soil, threw Ivan into the water just a few meters ahead. Then he struggled to get out of the mud and into the water himself. That was easier said then done however, because the impact of landing with their combined weights made his legs pretty well wedged into the mud.
Whether or not the amulet works on the mud depends on what the mud is. If it's possessed mud controlled by the demon, it would make sense for it to work. If it's the demon's body and it just happens to look and behave similarly to mud, then it wouldn't work.