Context 1:Our neighbour does this everyday in the morning to ensure that nothing (pleghm) remains in this throat. He pushes fingers into his throat and touchs his uvula. He provokes his uvula to vomit.
Context 2: I too do this once year; I drink salted water and forcibly vomit it.
You could use the phrase "inducing vomiting by putting his fingers down his throat" in order to clear his throat of phlegm. [Be aware that phlegm is mucus from the respiratory tract - so I'm not sure of the logic of inducing vomiting to clear one's throat].