Differential equations with discrete and distributed delays are considered. An explicit numerical Runge-Kutta method of order four is adapted for such equations. The problems of keeping the fourth order of convergence in practical implementation are studied. It is proposed to use local Hermitian interpolation of the third order to approximate the solution between time-mesh points to avoid interpolating over possible discontinuity points of the solution derivatives. The integral term is calculated using Simpson's method over the previously completed steps of Runge-Kutta method. The numerical solution of test problems confirms the declared fourth order of convergence of the constructed method.