This work is devoted to finding the optimal parameters of the HgTe/CdHgTe multi-quantum-well structure for the generation of two-dimensional plasmon-phonons under optical excitation. It is shown that a decrease in the bandgap and an increase in the number of quantum wells lead to two consequences. The first is a decrease in the nonequilibrium carrier threshold concentration and the threshold power density of optical excitation required for the plasmon-phonon gain. The second is a significant (tens of times) decrease in the effective refractive index of the generated plasmon-phonons, which improves the output of the plasmon-phonon radiation from the edge of the structure. The value of the optimal bandgap and the optimal number of quantum wells in the structure for plasmon-phonon gain are discussed.