This paper introduces and investigates a novel integrable inhomogeneous lattice that can be applied to the study of numerous nonlinear lattice equations affected by the spatiotemporally random force. Using the Tu scheme method, we first produce a completely new integrable lattice hierarchy containing the previously described equation and associated Lax pair. From there, we extract its Hamiltonian structures and numerous conservation laws. Employing the derived Lax pair, the discrete generalized \((n, N-n)\) -fold Darboux transformation with respect to this equation is then constructed for the first time. Then, using the resulting Darboux transformation, various localized waves like solitons, breathers, rational breathers, and mixed interaction structures are studied and illustrated visually, demonstrating that these wave structures are influenced by external potential coefficients. Specifically, these fresh localized waves with oscillating patterns of waves are not the same as the straight-line and spatiotemporal localization behavior of evolution and collision in the semi-discrete complex modified Korteweg-de Vries (cmKdV) system without external potential coefficients. Moreover, a number of significant physical quantities regarding two-soliton structures are precisely derived using the asymptotic analysis technique. Lastly, the numerical simulation is used to study dynamical behaviors with respect to numerous example localized waves. All of these waves are less impacted and robust by small perturbations according to numerical results. Our findings might be helpful in the interpretation of certain wave signal propagation phenomena that arise in domains like magnetism and electricity.
