The present study proposes and implements a high-fidelity data-driven emulation framework to predict spatiotemporal flow field in gas-centered, liquid-swirl coaxial (GCLSC) injectors, operating at supercritical conditions. In this injector, the mixing process of high-speed gaseous oxygen and swirling kerosene renders complicated flow dynamics. The emulation framework employs the common kernel-smoothed proper orthogonal decomposition (CKSPOD) technique as the surrogate model. The CKSPOD technique extracts dominant coherent flow structures through Hadamard-based POD, conducts kriging-based training process, and then reconstructs the structures to predict the flowfield of a new case. Significant improvements, including common-grid interpolation and physics-based conditions, are incorporated to the current framework to accommodate the prediction of complicated flow dynamics and mixing characteristics with varying geometry. In this study, recess length is chosen as the varying design parameter, and the LES results of GCLSC injectors at all design settings are collected to train the CKSPOD-based emulator. Detailed evaluations of the predicted flow fields are carried out, and the current framework can capture spatiotemporally evolving flow field, as well as propellant mixing with high accuracy. Good agreements are also achieved for time-averaged flow fields. In addition to the highly accurately prediction, the developed framework significantly reduces the computational time for the evaluation of new injector designs and will serve as a competitive tool for efficient survey of the design space.