Let $\mathcal{A}$ be a unital $\ast$-algebra containing a non-trivial projection. In this paper, we prove that if a map $\mathrm{\Omega }$ : $\mathcal{A}$ $\to$ $\mathcal{A}$ such that

$$\mathrm{\Omega }([\mathcal{K},\mathcal{F}{]}_{\ast }\odot \mathcal{D})=[\mathrm{\Omega }(\mathcal{K}),\mathcal{F}{]}_{\ast }\odot \mathcal{D}+[\mathcal{K},\mathrm{\Omega }(\mathcal{F}){]}_{\ast }\odot \mathcal{D}+[\mathcal{K},\mathcal{F}{]}_{\ast }\odot \mathrm{\Omega }(\mathcal{D}),$$

where $[\mathcal{K},\mathcal{F}{]}_{\ast }=\mathcal{K}\mathcal{F}-\mathcal{F}{\mathcal{K}}^{\ast }$ and $\mathcal{K}\odot \mathcal{F}={\mathcal{K}}^{\ast }\mathcal{F}+\mathcal{F}{\mathcal{K}}^{\ast }$ for all $\mathcal{K},\mathcal{F},\mathcal{D}\in \mathcal{A},$ then $\mathrm{\Omega }$ is an additive $\ast$-derivation. Furthermore, we extend its results on factor von Neumann algebras, standard operator algebras and prime $\ast$-algebras. Additionally, we provide an example illustrating the existence of such maps.