Viscoelastic materials are a kind of representative passive vibration control materials with many applications in civil engineering for earthquake mitigation in building structures, and these materials often serve in a thermo-elastic coupling environment. In this work, a one-dimensional magneto-thermoviscoelastic problem of a single-layer viscoelastic plate is investigated with memory-dependent derivative and nonlocal effect in the context of generalized thermo-elasticity. The plate is placed in a magnetic field, and the upper surface is subjected to a thermal shock. The governing equations for the single-layer plate are formulated considering the time delay and the kernel function of the memory-dependent derivative, nonlocal effect, temperature-dependent properties, and magnetic field. The Laplace transform and its numerical inversion are employed to solve this problem. The nondimensional temperature, displacement, and stress are calculated and presented graphically. Based on the numerical results, the influence of time delay and kernel function of the memory-dependent derivative, nonlocal effect parameters, temperature-dependent properties, and magnetic field parameters on the distributions of the nondimensional temperature, displacement, and stress are discussed.