  We provide a general result for bounding the difference between point
probabilities of integer supported distributions and the translated Poisson
distribution, a convenient alternative to the discretized normal. We illustrate
our theorem in the context of the Hoeffding combinatorial central limit theorem
with integer valued summands, of the number of isolated vertices in an
Erd\H{o}s-R\'enyi random graph, and of the Curie-Weiss model of magnetism,
where we provide optimal or near optimal rates of convergence in the local
limit metric. In the Hoeffding example, even the discrete normal approximation
bounds seem to be new. The general result follows from Stein's method, and
requires a new bound on the Stein solution for the Poisson distribution, which
is of general interest.
