  We show that any sequence $(x_n)_{n \in \mathbb{N}} \subseteq [0,1]$ that has
Poissonian correlations of $k$-th order is uniformly distributed, also
providing a quantitative description of this phenomenon. Additionally, we
extend connections between metric correlations and additive energy, already
known for pair correlations, to higher orders. Furthermore, we examine how the
property of Poissonian $k$-th correlations is reflected in the asymptotic size
of the moments of the function $F(t,s,N) = #\{n\leq N : \|x_n - t\| \leq s/(2N)
\},\, t\in [0,1]. $
