The prediction-correction framework developed in [B. He, Splitting Contraction Algorithm for Convex Optimization, Science Press, 2025] is a simple yet powerful tool for analyzing the convergence of diverse first-order optimization methods, including the Augmented Lagrangian Method (ALM) and the Alternating Direction Method of Multipliers (ADMM). In this paper, we propose a generalized prediction-correction framework featuring a parameter-free relaxation iteration which is a key enhancement over existing frameworks. Leveraging variational characterizations of the first-order optimality conditions for each subproblem, we establish its global convergence and sublinear convergence rates in both ergodic and nonergodic senses. Furthermore, this new framework is applied to reformulate an indefinite linearized ALM, the Chambolle-Pock method, and several ADMM-type methods, enabling concise analyses of their convergence conditions. These results demonstrate the framework’s versatility in unifying and simplifying convergence analyses for a broader class of first-order methods.