  We study chiral anomalies in $\mathcal N=(0, 1)$ and $(0, 2)$ two-dimensional
minimal sigma models defined on generic homogeneous spaces $G/H$. Such minimal
theories contain only (left) chiral fermions and in certain cases are
inconsistent because of "incurable" anomalies. We explicitly calculate the
anomalous fermionic effective action and show how to remedy it by adding a
series of local counter-terms. In this procedure, we derive a local anomaly
matching condition, which is demonstrated to be equivalent to the well-known
global topological constraint on $p_1(G/H)$. More importantly, we show that
these local counter-terms further modify and constrain "curable" chiral models,
some of which, for example, flow to nontrivial infrared superconformal fixed
point. Finally, we also observe an interesting relation between $\mathcal N=(0,
1)$ and $(0, 2)$ two-dimensional minimal sigma models and supersymmetric gauge
theories.
  This paper generalizes and extends the results of our previous publication
arXiv:1510.04324.
