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We study the superradiance amplification factor (SAF) for a charged massive scalar wave scattering off small and slowly rotating Kerr-Newman black holes in $f(R)$ gravity immersed in asymptotically flat and de-Sitter spacetimes. We employ the \emph{``analytical asymptotic matching''} approximation technique which is valid in low frequency regime where the Compton wavelength of the propagating particle is much larger than the size of the black hole. The $f(R)$-Kerr-Newman family solution induces an extra distinguishable effect on the contribution of the black hole's electric charge to the metric and that in turn affects the SAFs and their frequency ranges. While our analysis are general, we present the numerical results for the Starobinsky and Hu-Sawicki $f(R)$ models of gravity as our working examples. In the case of asymptotically flat spacetime, the SAFs predicted in Starobinsky $f(R)$ model are not distinguishable from those of GR while for Hu-Sawicki model the SAFs can be weaker or stronger than those of GR within the frequency parameters space. In the case of asymptotically de-Sitter spacetime, the superradiance scattering may not either occur in Starobinsky model or has a weaker chance compared to GR while in Hu-Sawicki model the results of SAFs and their frequency regimes are different from the standard ones.