The discussion is devoted to the evaluation of the radiative one-loop correction with a vertex, where the vertex represents the interaction with an external magnetic field, and the loop refers to a "ordinary" or thermal photon. Within the framework of the rigorous QED theory, corrections to the g-factor of the bound electron and the level width are derived as real and imaginary parts, respectively. The effects are considered for hydrogen and light hydrogen-like ions and are discussed in regard to modern experiments in this field. Temperatures and magnetic field strengths corresponding to various laboratory and astrophysical conditions are taken into account. The discussion is based on our most recent results obtained within the QED approach at the finite temperature being developed by our group.