Building on previous efforts dedicated to the eduction of flow motions associated with low-angle sound emission from turbulent jets, the current paper proposes a strategy for the reduced-order dynamical modelling of these motions. The technique is data-driven and comprises (i) a dynamic Ansatz obtained by Galerkin projection of the Navier-Stokes equations onto a set of POD orthonormal spatial basis functions coupled with (ii) a dual-Kalman filter which allows long term prediction of the state and correction in line of the model's parameters. Rather than directly computing the coefficients of the model equation driving the flow state, which is only possible when full flow information is available, the coefficients are first identified thanks to a regularised L2 minimisation procedure. A twenty degree-of-freedom reduced-order dynamical model (ROM) is then established. The identification scheme is found to lead to bounded ROMs but with estimated trajectories drifting from the true ones. A data-driven correction based on a dualensemble Kalman filter is therefore implemented and an empirical non-linear eddy viscosity model is added to the ROM to solve its inherent lack of stability. This correction allows the estimated state to be corrected "on-line" when a measurement of the true state (here point velocity and far-field pressure) is available, as well as to find the optimal set of parameters associated with the eddy viscosity term. The described approach, though validated here using partial flow information from a numerical simulation, has been designed in the perspective of being implemented in an experimental context.