  A new symmetry of $(1,0)$ supersymmetric non-linear $\sigma$-models in two
dimensions with Fermi and mass sectors is introduced. It is a generalisation of
the so-called special holonomy $W$-symmetry of Howe and Papadopoulos associated
with structure group reductions of the target space $\mathcal{M}$. Our symmetry
allows in particular non-trivial flux and instanton-like connections on vector
bundles over $\mathcal{M}$. We also investigate potential anomalies and show
that cohomologically non-trivial terms in the quantum effective action are
invariant under a corrected version of our symmetry. Consistency with heterotic
supergravity at first order in $\alpha'$ is manifest and discussed.
