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Manganese telluride (MnTe) is a prospective platform for ultrafast carrier dynamics, spin-based thermoelectrics, and magnon-drag transport due to its unique electronic and magnetic properties. We use inelastic neutron scattering to study both pure and lithium-doped MnTe, focusing on the influence of doping in opening a magnon gap. We use neutron powder diffraction to determine critical exponents for the phase transition in both pure and Li-doped MnTe and complement this information with muon spin rotation/relaxation. The opening of the magnon gap and spin reorientation in Li-doped MnTe is mainly due to increased magnetic anisotropy along the [001] axis, a feature not present in pure MnTe.