Quantum systems face inherent challenges in achieving precise control, and solving the Schr"{o}dinger equation is often intractable for complex hybrid platforms. Here, we introduce a magnon into a time-delayed feedback-controlled quantum system. To solve the dynamics numerically and efficiently, we employ a Long Short-Term Memory network, a machine learning approach, to propagate the probability amplitudes to the steady state. By applying feedback, we effectively stabilize the intracavity state. Our results reveal that the photon-photon correlation function and the concurrence, a measure of magnon-atom entanglement, exhibit periodic oscillations with the cavity-mirror distance, and that feedback significantly enhances both antibunching and bunching when the detuning is varied. These findings not only demonstrate the power of artificial intelligence in quantum dynamics simulation, but also open a promising route for on-demand quantum state engineering in hybrid magnonic systems, with potential applications in quantum networks and quantum information processing.