This paper is concerned with the existence and the limiting behavior of measure attractors of distribution laws of the solution segment process for the McKean-Vlasov stochastic $p$-Laplace lattice system with time delay driven by Lévy noise. The nonlinear drift and diffusion terms are allowed to have superlinear growth. Due to time delay, the Skorohod metric space is employed to describe the trajectories of the solutions with jumps. We first prove the existence and uniqueness of càdlàg solutions for the lattice system, and then define a non-autonomous cocycle acting on the Borel probability measures in the Skorohod space. This cocycle is continuous in bounded subsets of the space of probability measures only when time is sufficiently large. We then prove the existence of pullback absorbing sets and the asymptotic compactness of the cocycle as well as the existence and uniqueness of pullback measure attractors. We finally investigate the limiting behavior of measure attractors of the lattice system as the noise intensity approaches zero, and establish the optimal convergence rate of singleton measure attractors in the Wasserstein distance of order $θ$.
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