  We show that the combined effect of cosmic magnetic field and a possible
non-standard interaction between baryons and dark matter has interesting
consequences on the thermal Sunyaev$-$Zel${'}$dovich (tSZ) effect depending on
the temperature and the ionization state of the intergalactic medium. The drag
force between the baryons and dark matter due to the relative velocity between
them, and their temperature difference results in heat transfer between these
two species. At the same time the ambipolar diffusion and the decaying magnetic
turbulence tends to heat up the baryons. This interplay of these two processes
give rise to different evolution histories of the thermal and ionization state
of the universe and hence influences the CMB spectrum at small scales through
the tSZ effect. In this work, we have computed the evolution of the
temperature, ionization fraction and the y-parameter of the CMB for different
strengths of the magnetic field and the interaction cross-section. We note that
an interaction cross-section of $\sigma_0=10^{-42}$ m$^{2}$ (with a magnetic
field strength of $3.0$ nG) reduces the y-parameter by an order of magnitude as
compared to the case with similar strength of magnetic field but where there is
no such interaction between the baryons and dark matter.
