The lack of radiation sources in the frequency range of 7-10 THz is associated with strong absorption of the THz waves on optical phonons within the GaAs Reststrahlen band. To avoid such absorption, we propose to use HgCdTe as an alternative material for THz quantum cascade lasers thanks to a lower phonon energy than in III-V semiconductors. In this work, HgCdTe-based quantum cascade lasers operating in the GaAs phonon Reststrahlen band with a target frequency of 8.3 THz have been theoretically investigated using the balance equation method. The optimized active region designs, which are based on three and two quantum wells, exhibit the peak gain exceeding 100 cm-1 at 150 K. We have analyzed the temperature dependence of the peak gain and predicted the maximum operating temperatures of 170 K and 225 K for three- and two-well designs, respectively. At temperatures exceeding 120 K, the better temperature performance has been obtained for the two-well design, which is associated with a larger spatial overlap of weakly localized lasing wavefunctions, as well as, a higher population inversion. We believe that the findings of this work can open a pathway towards the development of THz quantum cascade lasers featuring a high level of optical gain due to the low electron effective mass in HgCdTe.