Modeling studies were carried out to examine the thermophoretic magnetohydrodynamics flow phenomena in a penetrable wedge with the higher-order (second-order) slip interaction. The model problem including mass, momentum, and energy equations transforms into nonlinear ordinary differential equations by exploiting a new class of similarity approach. The modified model equations are solved using the Nachtsheim-Swigert shooting method with the sixth-order Runge-Kutta integration scheme. The numerical outputs generated for the stream function, velocity, and the local skin friction are equated with the past results presented in the literature and found to have excellent precision. The modeling results reveal that the velocity outlines drop gradually with the lessening of different model constraints comprising mass transfer coefficient, wedge angle, Prandtl number, Schmidt number, first-order slip parameters, and shrinkages with the intensification of second-order slip constraint. The outcomes also show that the temperature profiles rise with the upsurge of Biot number, and Schmidt number, whereas the reverse scenarios are observed for Hartmann number, and unsteadiness parameter. Numerical results for the effects of the distinct constraints named the local skin-friction coefficient, rate of heat and mass transfer, thermophoretic velocity, and thermophoretic deposition velocity are also exposed in tabular form and discussed in detail.