  We obtain Berry-Esseen-type bounds for the sum of random variables with a
dependency graph and uniformly bounded moments of order $\delta \in (2,\infty]$
using a Fourier transform approach. Our bounds improve the state-of-the-art in
the regime where the degree of the dependency graph is large. As a Corollary of
our results, we obtain a Central Limit Theorem for random variables with a
sparse dependency graph that are uniformly bounded in $L^{\delta}$ for some
$\delta\in(2,\infty]$.
