The coherent light-matter interaction has drawn an enormous amount of attention for its fundamental importance in the cavity quantum-electrodynamics (C-QED) field and great potential in quantum information applications. Here, we design a hybrid C-QED system consisting of a quantum dot (QD) driven by two-tone fields implanted in a photonic crystal (PhC) cavity coupled to an auxiliary cavity with a single-mode waveguide and investigate the hybrid system operating in the weak, intermediate, and strong coupling regimes of the light-matter interaction via comparing the QD-photon interaction with the dipole decay rate and the cavity field decay rate. The results indicate that the auxiliary cavity plays a key role in the hybrid system, which affords a quantum channel to influence the absorption of the probe field. By controlling the coupling strength between the auxiliary cavity and the PhC cavity, the phenomenon of the Mollow triplet can appear in the intermediate coupling regime, and even in the weak coupling regime. We further study the strong coupling interaction manifested by vacuum Rabi splitting in the absorption with manipulating the cavity-cavity coupling under different parameter regimes. This study provides a promising platform for understanding the dynamics of QD-C-QED systems and paving the way toward on-chip QD-based nanophotonic devices.