The effect of the familiar Rayleigh mechanism of energy release in an elastic medium (which plays an important role, in particular, in gas discharge plasma) on the structure of a running shock wave (SW) is considered in the general case in the 1D approximation. The equation describing the propagation of the SW in this case is derived. An analytic solution to this equation is obtained for small values of the parameter characterizing the properties of the medium. The type of the solution for different signs of this parameter and for its values modulo equal to unity is analyzed. It is found that, for positive values of this parameter, a SW in the form of a step is suppressed in such a medium and degenerated into a perturbation in the form of a hump. On the contrary, for negative values of the parameter, the SW is enhanced. It is found that a stationary solution exists in the system of coordinates associated with the SW propagation in a medium with the Rayleigh energy release mechanism only if the boundary of the medium lies downstream from the shock layer. The position of this boundary corresponds to the so-called critical energy supply and the local Mach number is equal to unity at this point. For a positive value of the parameter of the medium with the Rayleigh energy release mechanism, the equation of propagation has no stationary solution for any position of the boundary of the medium upstream from the shock layer when the value of the parameter exceeds a certain limiting value. The results make it possible to analyze the features of SW propagation in a weakly ionized gas discharge plasma.