"Mortals, be wiped out! Everything... fall beneath Zamasu!"
"Half-Corrupted" is an "ugly form" utilized by Fused Zamasu in the anime, turning him into a mutated state. In this form, Fused Zamasu is known as "Grotesque" Fused Zamasu. In the anime, the state is attained due to receiving extensive damage and further powering up through the Light of Justice technique. In the manga, the state is attained when the Potara Fusion ran out, but due to Future Zamasu and Goku Black being one and the same, their synergy caused them to reject the fission.
Fused Zamasu is created from the fusion of Future Zamasu, who is immortal, and Goku Black, who isn't immortal and is far stronger than Future Zamasu. Because of this Fused Zamasu was not completely immortal and became half corrupted.
After having half of his body disfigured by Goku's God Kamehameha, Fused Zamasu calls upon the Light of Justice to severely injure his arm - causing his arm to become larger, much more muscular and creating more scarred flesh - but also increased Fused Zamasu's power. His aura also changed into a fusion of Future Zamasu's purple aura on the inside and the dark red outer part of Super Saiyan Rosé Goku Black's aura.
This increase allowed Fused Zamasu to power up even further, which forced Goku and Vegeta to fuse into Vegito to even the odds. During his fight with Vegito, Fused Zamasu's body becomes more and more unstable as his power and anger increases. Eventually, Fused Zamasu enters his Huge Final Stage in the climax of his fight with Vegito.
In the manga however, this state is completely different, and attained due to Fused Zamasu's immortality attempting to compensate the loss of cells in Fused Zamasu's body, due to Goku Black's and Future Zamasu's time fused being up. This was due to Goku Back and Future Zamasu being one and the same, and Fused Zamasu's immortality remaining fully intact in the fusion process, unlike the anime. However, this Half-Corrupted form was only a temporary state. As when Future Trunks cut Fused Zamasu in half, Goku Black and Future Zamasu soon remerged into Infinite Zamasu, who is identical to Fused Zamasu in appearance. After this, Fused Zamasu's immortality in his Infinite Zamasu state seemed to have found a solution in remerging the two Zamases together.
Video game appearances
This Half-Corrupted form of Zamasu debuts in Dokkan Battle. It later appears in the Dragon Ball Super Pack 4 of Dragon Ball Xenoverse 2. Zamasu in this form appears only in the story mode. It appears in two cutscenes, one where Vegito fights Fused Zamasu, and a cinematic cutscene where Vegito eventually defuses and then is defeated by Super Saiyan Anger Future Trunks' Final Hope Slash.