The prediction-correction framework developed in [B. He, Splitting Contraction Algorithm for Convex Optimization, Science Press, 2025] is a simple yet practical technique to analyze the convergence of many first-order methods, such as the Augmented Lagrangian Method (ALM), the Alternating Direction Method of Multipliers (ADMM) and so on. In this paper, we propose a generalized prediction-correction framework which enjoys a flexible parameter-free forward/backward iteration. We establish its global convergence and sublinear convergence rate in the ergodic/nonergodic sense by the aid of variational characterization for the first-order optimality condition of each subproblem. This novel prediction-correction framework is also applied to reformulate an indefinite linearized ALM, the Chambolle-Pock method and some ADMM-type methods with concise analysis on their convergence conditions.