  We determine the general structure of quantum anomalies for the $R$-multiplet
of four dimensional $\mathcal{N}=1$ supersymmetric quantum field theories in
the presence of background fields for an arbitrary number of Abelian flavor
multiplets. By solving the Wess-Zumino consistency conditions for off-shell new
minimal supergravity in four dimensions with an arbitrary number of Abelian
vector multiplets, we compute the anomaly in the conservation of the
supercurrent to leading non trivial order in the gravitino and vector multiplet
fermions. We find that both $R$-symmetry and flavor anomalies necessarily lead
to a supersymmetry anomaly, thus generalizing our earlier results to non
superconformal theories with Abelian flavor symmetries. The anomaly in the
conservation of the supercurrent leads to an anomalous transformation for the
supercurrent under rigid supersymmetry on bosonic backgrounds that admit new
minimal Killing spinors. The resulting deformation of the supersymmetry algebra
has implications for supersymmetric localization computations on such
backgrounds.
