At present, hydrogen-based compounds constitute one of the most promising classes of materials for applications as a phonon-mediated high-temperature superconductors. Herein, the behavior of the superconducting phase in tellurium hydride (HTe) at high pressure ($p=300$ GPa) is analyzed in details, by using the isotropic Migdal-Eliashberg equations. The chosen pressure conditions are considered here as a case study which corresponds to the highest critical temperature value ($T_{c}$) in the analyzed material, as determined within recent density functional theory simulations. It is found that the Migdal-Eliashberg formalism, which constitutes a strong-coupling generalization of the Bardeen-Cooper-Schrieffer (BCS) theory, predicts that the critical temperature value ($T_{c}=52.73$ K) is higher than previous estimates of the McMillan formula. Further investigations show that the characteristic dimensionless ratios for the the thermodynamic critical field, the specific heat for the superconducting state, and the superconducting band gap exceeds the limits of the BCS theory. In this context, also the effective electron mass is not equal to the bare electron mass as provided by the BCS theory. On the basis of these findings it is predicted that the strong-coupling and retardation effects play pivotal role in the superconducting phase of HTe at 300 GPa, in agreement with similar theoretical estimates for the sibling hydrogen and hydrogen-based compounds. Hence, it is suggested that the superconducting state in HTe cannot be properly described within the mean-field picture of the BCS theory.
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