  In this article we establish for the superdiffusive regime $p \in (1/2,1)$
that the fluctuations of a general step-reinforced random walk around $a_n
\hat{W}$, where $(a_n)_{n \in \mathbb{N}}$ is a non-negative sequence of order
$n^p$ and $\hat{W}$ is a non-degenerate random variable, is Gaussian. This
extends a known result by Kubota and Takei for the elephant random walk to the
more general setting of step-reinforced random walks. Further, we provide an
application of the asymptotic normality of $\hat{S}$ around $a_n \hat{W}$ to
reinforced empirical processes as studied recently by Bertoin, which yields a
refined Donsker's invariance principle.
