In this paper we investigate $\Delta$-weak character amenability of certain Banach algebras such as projective tensor product $A\widehat{\otimes }B$ and Lau product $A{\times }_{\theta }B$, where $A$ and $B$ are two arbitrary Banach algebras and $\theta \in \Delta \left(B\right)$, the character space of $B$. We also investigate $\Delta$-weak character amenability of abstract Segal algebras and module extension Banach algebras.