The fabrication of integrated quantum dot (QD)-optical microcavity systems is a requisite step for the realization of a wide range of nanophotonic experiments (and applications) that exploit the ability of QDs to emit nonclassical light, e.g., single photons. Thanks to their \ensuremath{\sim}20-nm positioning accuracy and to their proven potential for single-photon operation, the QDs obtained by spatially selective hydrogen irradiation of dilute-nitride semiconductors ---such as Ga(AsN) and Ga(PN)---are uniquely suited for integration with photonic nanodevices. In the present work, we demonstrate the ability to deterministically integrate single, site-controlled Ga(AsN)/Ga(AsN):H QDs within a photonic crystal (PhC) cavity. The properties of the fabricated QD-PhC cavity systems are then probed by photon correlation---providing clear evidence of single-photon emission---and time-resolved microphotoluminescence spectroscopy. Detailed information on the dynamics of our integrated nanodevices can be inferred by comparing these experiments to the solutions of a rate-equations system, developed by taking into account all the main processes leading to the capture, relaxation, and recombination of carriers in and out of the QD. This allows us to follow the evolution of the relevant recombination rates in our system for varying energy detuning, \ensuremath{\Delta}$E$, between the QD and the PhC cavity. When the QD exciton transition is nearly resonant with the cavity mode, a large (gtenfold) enhancement of the spontaneous emission rate is observed, in substantial agreement with Jaynes-Cummings (JC) theory. For intermediate detunings (\ensuremath{\Delta}$E$ \ensuremath{\sim} 1.5--3.5 meV), on the other hand, the observed enhancement is significantly larger than that predicted by JC theory, due to the important role played by acoustic phonons in mediating the QD-PhC cavity coupling in a solid-state environment. Apart from its fundamental interest, the observation of such phonon-mediated, broadband enhancement of light-matter interaction significantly relaxes the requirements for the realization of a large variety of cavity QED-based experiments and applications. These include many photonic devices for which the use of site-controlled Ga(AsN)/Ga(AsN):H QDs would be inherently advantageous, such as those based on the coupling between more than one QD and a single cavity mode (e.g., few-QD nanolasers and QD solids).