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A new symmetry of $(1,0)$ supersymmetric non-linear $\sigma$-models in twodimensions with Fermi and mass sectors is introduced. It is a generalisation ofthe so-called special holonomy $W$-symmetry of Howe and Papadopoulos associatedwith structure group reductions of the target space $\mathcal{M}$. Our symmetryallows in particular non-trivial flux and instanton-like connections on vectorbundles over $\mathcal{M}$. We also investigate potential anomalies and showthat cohomologically non-trivial terms in the quantum effective action areinvariant under a corrected version of our symmetry. Consistency with heteroticsupergravity at first order in $\alpha'$ is manifest and discussed.

$\mathcal{G}$-structure symmetries and anomalies in $(1,0)$ non-linear $\sigma$-models