  Consider a branching random walk, where the branching mechanism is governed
by a Galton-Watson process, and the migration by a finite range symmetric
irreducible random walk on the integer lattice $\mathbb{Z}^d$. Let $Z_n(z)$ be
the number of the particles in the $n$-th generation at the point $z\in
\mathbb{Z}^d$. Under the mild moment conditions for offspring distribution of
the underlying Galton-Watson, we derive a second order expansion in the local
limit theorem for $Z_n(z)$ for each given $z\in \mathbb{Z}^d$. That generalizes
the results for simple branching random walks obtained by Gao [2018, SPA].
