<p>Stay cables can exhibit undesirable vibrations under external excitations. This paper studies the vibration control of a stay cable structure via a time-delayed nonlinear energy sink with nonlinear damping. The system dynamics are modeled separately for uniformly distributed load and support motion. The Galerkin method is used to discretize the original partial differential equations. The dynamic responses of the system under the two cases are obtained. The reasonable intervals for the control parameters are determined through a bifurcation analysis of the trivial solution. To elucidate the parametric influence, the effects of nonlinear damping, excitation amplitude, and frequency on the displacement amplitude are quantified through numerical analysis. The influences of the control parameters on the dynamic responses and energy of the system are explored. The genetic algorithm for parameter optimization is employed and the controller exhibits high performance with tunable parameters. The bifurcation diagram is given and the complicated dynamical features are observed. These results demonstrate that time-delayed feedback control plays a key role in improving the vibration reduction performance of nonlinear energy sink for cable system vibration mitigation, with specific manifestations including the suppression of resonant peaks and the alleviation of complicated dynamic behaviors in the stay cable. The novelty of this work lies in leveraging the interplay between these nonlinear elements to control the dynamics of the stay cable.</p>