Dynamic models of large-deformation structures are widely used to simulate the transient response of large-scale and lightweight structures, but their geometric nonlinearity poses significant challenges for efficient and accurate modeling and control. The two-field reduced-order model (ROM) for large-deformation structures can be constructed, via data-driven Proper Orthogonal Decomposition (POD) with the Hellinger-Reissner variational principle. This model incorporates prior stress information and reduces the order of stiffness invariants derived from displacement bases from fourth to third order, thereby improving both the accuracy and efficiency of transient dynamic model reduction. Based on the two-field ROM, this work further investigates efficient prediction and optimal control of flexible structural responses by embedding the ROM into a model predictive control (MPC) framework to achieve trajectory optimization and tracking. Case studies on a slender beam system and a wing skeleton demonstrate that the proposed approach significantly reduces online computational cost while maintaining accuracy, and outperforms traditional displacement-based POD methods in both efficiency and accuracy, showing strong applicability to MPC-based control of nonlinear flexible structures.