  This paper is concerned with the existence of metrics of constant Hermitian
scalar curvature on almost-K\"ahler manifolds obtained as smoothings of a
constant scalar curvature K\"ahler orbifold, with $A_1$ singularities. More
precisely, given such an orbifold that does not admit nontrivial holomorphic
vector fields, we show that an almost-K\"ahler smoothing $(M_\varepsilon,
\omega_\varepsilon)$ admits an almost-K\"ahler structure $(\hat J_\varepsilon,
\hat g_\varepsilon)$ of constant Hermitian curvature. Moreover, we show that
for $\varepsilon>0$ small enough, the $(M_\varepsilon, \omega_\varepsilon)$ are
all symplectically equivalent to a fixed symplectic manifold $(\hat M, \hat
\omega)$ in which there is a surface $S$ homologous to a 2-sphere, such that
$[S]$ is a vanishing cycle that admits a representant that is Hamiltonian
stationary for $\hat g_\varepsilon$.
