  We present an accurate non-linear matter power spectrum prediction scheme for
a variety of extensions to the standard cosmological paradigm, which uses the
tuned halo model previously developed in Mead (2015b). We consider dark energy
models that are both minimally and non-minimally coupled, massive neutrinos and
modified gravitational forces with chameleon and Vainshtein screening
mechanisms. In all cases we compare halo-model power spectra to measurements
from high-resolution simulations. We show that the tuned halo model method can
predict the non-linear matter power spectrum measured from simulations of
parameterised $w(a)$ dark energy models at the few per cent level for
$k<10\,h\mathrm{Mpc}^{-1}$, and we present theoretically motivated extensions
to cover non-minimally coupled scalar fields, massive neutrinos and Vainshtein
screened modified gravity models that result in few per cent accurate power
spectra for $k<10\,h\mathrm{Mpc}^{-1}$. For chameleon screened models we
achieve only 10 per cent accuracy for the same range of scales. Finally, we use
our halo model to investigate degeneracies between different extensions to the
standard cosmological model, finding that the impact of baryonic feedback on
the non-linear matter power spectrum can be considered independently of
modified gravity or massive neutrino extensions. In contrast, considering the
impact of modified gravity and massive neutrinos independently results in
biased estimates of power at the level of 5 per cent at scales
$k>0.5\,h\mathrm{Mpc}^{-1}$. An updated version of our publicly available
HMcode can be found at https://github.com/alexander-mead/HMcode
