  We introduce $S_z$ spin-projection based on cluster mean-field theory and
apply it to the ground state of strongly-correlated spin systems. In cluster
mean-field, the ground state wavefunction is written as a factorized tensor
product of optimized cluster states. In previous work, we have focused on
unrestricted cluster mean-field, where each cluster is $S_z$ symmetry adapted.
We here remove this restriction by introducing a generalized cluster mean-field
(GcMF) theory, where each cluster is allowed to access all $S_z$ sectors,
breaking $S_z$ symmetry. In addition, a projection scheme is used to restore
global $S_z$, which gives rise to $S_z$ spin-projected generalized cluster
mean-field (S$_z$GcMF). Both of these extensions contribute to accounting for
inter-cluster correlations. We benchmark these methods on the 1D, quasi-2D, and
2D $J_1-J_2$ and $XXZ$ Heisenberg models. Our results indicate that the new
methods (GcMF and S$_z$GcMF) provide a qualitative and semi-quantitative
description of the Heisenberg lattices in the regimes considered, suggesting
them as useful references for further inter-cluster correlations, which are
discussed in this work.
