The first stars in the Universe, the so-called Population III stars, form in small dark matter minihaloes with virial temperatures $T_{rm vir} < 10^{4}$~K. Cooling in these minihaloes is dominated by molecular hydrogen (H$_{2}$), and so Population III star formation is only possible in those minihaloes that form enough H$_{2}$ to cool on a short timescale. As H$_{2}$ cooling is more effective in more massive minihaloes, there is therefore a critical halo mass scale $M_{rm min}$ above which Population III star formation first becomes possible. Two important processes can alter this minimum mass scale: streaming of baryons relative to the dark matter and the photodissociation of H$_{2}$ by a high redshift Lyman-Werner (LW) background. In this paper, we present results from a set of high resolution cosmological simulations that examine the impact of these processes on $M_{rm min}$ and on $M_{rm ave}$ (the average minihalo mass for star formation), both individually and in combination. We show that streaming has a bigger impact on $M_{rm min}$ than the LW background, but also that both effects are additive. We also provide fitting functions quantifying the dependence of $M_{rm ave}$ and $M_{rm min}$ on the streaming velocity and the strength of the LW background.
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