  The centred pyrochlore lattice is a novel geometrically frustrated lattice,
realized in the metal-organic framework Mn(ta)$_2$ (arXiv:2203.08780) where the
basic unit of spins is a five site centred tetrahedron. Here, we present an
in-depth theoretical study of the $J_1-J_2$ classical Heisenberg model on this
lattice, using a combination of mean-field analytical methods and Monte Carlo
simulations. We find a rich phase diagram with low temperature states
exhibiting ferrimagnetic order, partial ordering, and a highly degenerate spin
liquid with distinct regimes of low temperature correlations. We discuss in
detail how the regime displaying broadened pinch points in its spin structure
factor is consistent with an effective description in terms of a fluid of
interacting charges. We also show how this picture holds in two dimensions on
the analogous centred kagome lattice and elucidate the connection to the
physics of thin films in ($d+1$) dimensions. Furthermore, we show that a
Coulomb phase can be stabilized on the centred pyrochlore lattice by the
addition of further neighbour couplings. This demonstrates the centred
pyrochlore lattice is an experimentally relevant geometry which naturally hosts
emergent gauge fields in the presence of charges at low energies.
