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The objective of this work is to improve an appropriate generalized thermoelastic heat transport framework. In the proposed model, the mathematical heat transfer equation is characterized by the fourth-order Moore-Gibson-Thompson (MGT) equation. Construction of the system equation in Green-Naghdi type III model involves the addition of the phase lag delay coefficient up to the second degree. This proposed model is advantageous because it is compatible with observable physical processes and allows speed reduction as heat waves travel within the solid. The presented model can also be used to derive a number of alternative models of thermoelasticity as special cases. The Hall current influence is considered to analyze the magneto-thermoelastic couplings in an infinite conducting viscoelastic medium with a cylindrical cavity under a strong magnetic field propagating along the cavity axis. It is assumed that the viscoelastic material of the medium is Kelvin-Voigt type. Also, in contrast to many cases where the thermal conductivity factor remains unchanged, it is considered that this coefficient varies with changes in temperature. To solve the system of equations, the Laplace transform methodology is used. The studied fields are shown schematically, and the implications of viscosity and thermal fluctuation are explained. Moreover, tabular representations of numerical data are provided, and the suggested model is validated via comparison with the existing frameworks. • Development of a proper generalized thermoelastic heat transfer framework. • The fourth-order Moore-Gibson-Thompson (MGT) equation is used. • The Hall current influence is considered for the magneto-thermoelastic couplings. • Laplace transform method is employed to solve the system of equations. • Thermal conductivity factor varies with changes in temperature. [ABSTRACT FROM AUTHOR]