The Volterra-Hammerstein integral equation plays a crucial role in the control of robotic manipulators, which are widely used in industrial automation, medical robotics, and space exploration. These systems present significant control challenges due to their nonlinear behaviors and memory-dependent effects. In this research, we establish novel common fixed point theorems for generalized rational contractions within the framework of complex-valued suprametric spaces. Leveraging these theoretical advancements, we apply the derived results to solve the Volterra-Hammerstein integral equation, demonstrating its significance in robotic manipulator control. To emphasize the originality and practical applicability of our findings, a comprehensive illustrative example is provided.