<p>This paper investigates the stabilization of a transport PDE-ODE cascade system where the transport speed is influenced by the integral of the ODE states and the boundary state of the PDE. The presence of this integral term complicates the actuator dynamics, making traditional predictor methods and conventional feedback control designs ineffective. To address this challenge, we propose a novel bilayer predictor-feedback control strategy, which accurately predicts the actuator dynamics and compensates for the transport delay. The global asymptotic stability of the closed-loop system is established using backstepping and Lyapunov methods. Furthermore, we provide an equivalent representation of the PDE-ODE cascade as a nonlinear system with an input delay implicitly defined by an integral of past ODE states and delayed inputs. A corresponding predictor-feedback control law is also formulated for this delay system. Numerical example is showcased to validate the effectiveness of our control design.</p>