  We investigate the MSHT20 global PDF sets, demonstrating the effects of
varying the strong coupling $\alpha_S(M_Z^2)$ and the masses of the charm and
bottom quarks. We determine the preferred value, and accompanying
uncertainties, when we allow $\alpha_S(M_Z^2)$ to be a free parameter in the
MSHT20 global analyses of deep-inelastic and related hard scattering data, at
both NLO and NNLO in QCD perturbation theory. We also study the constraints on
$\alpha_S(M_Z^2)$ which come from the individual data sets in the global fit by
repeating the NNLO and NLO global analyses at various fixed values of
$\alpha_S(M_Z^2)$, spanning the range $\alpha_S(M_Z^2)=0.108$ to $0.130$ in
units of $0.001$. We make all resulting PDFs sets available. We find that the
best fit values are $\alpha_S(M_Z^2)=0.1203\pm 0.0015$ and $0.1174\pm 0.0013$
at NLO and NNLO respectively. We investigate the relationship between the
variations in $\alpha_S(M_Z^2)$ and the uncertainties on the PDFs, and
illustrate this by calculating the cross sections for key processes at the LHC.
We also perform fits where we allow the heavy quark masses $m_c$ and $m_b$ to
vary away from their default values and make PDF sets available in steps of
$\Delta m_c =0.05~{\rm GeV}$ and $\Delta m_b =0.25~{\rm GeV}$, using the pole
mass definition of the quark masses. As for varying $\alpha_S(M_Z^2)$ values,
we present the variation in the PDFs and in the predictions. We examine the
comparison to data, particularly the HERA data on charm and bottom cross
sections and note that our default values are very largely compatible with best
fits to data. We provide PDF sets with 3 and 4 active quark flavours, as well
as the standard value of 5 flavours.
