This article investigates Wardowski's contraction in the setting of extended distance spaces known as $M$-metric spaces using hybrid operators based an $M$ -dynamic iterative process. The main purpose is to observe new set-valued Chatterjea type common fixed point theorems for hybrid operators with respect to an $M$-dynamic iterative process, i.e., $\check{D}({\mathrm{\Psi }}_1,{\mathrm{\Psi }}_2,s_0)$. We realize an application: the existence of a solution for a multistage system and integral equation. Also, we give a graphical interpretation of our obtained theorems. The main results are explicated with the help of a relevant example. Some important corollaries are extracted from the main theorems as well.