  One-rank perturbations of Wigner matrices have been closely studied: let
$P=\frac{1}{\sqrt{n}}A+\theta vv^T$ with $A=(a_{ij})_{1 \leq i,j \leq n} \in
\mathbb{R}^{n \times n}$ symmetric, $(a_{ij})_{1 \leq i \leq j \leq n}$ i.i.d.
with centered standard normal distributions, and $\theta>0, v \in
\mathbb{S}^{n-1}.$ It is well known $\lambda_1(P),$ the largest eigenvalue of
$P,$ has a phase transition at $\theta_0=1:$ when $\theta \leq 1,$
$\lambda_1(P) \xrightarrow[]{a.s.} 2,$ whereas for $\theta> 1,$ $\lambda_1(P)
\xrightarrow[]{a.s.} \theta+\theta^{-1}.$ Under more general conditions, the
limiting behavior of $\lambda_1(P),$ appropriately normalized, has also been
established: it is normal if $||v||_{\infty}=o(1),$ or the convolution of the
law of $a_{11}$ and a Gaussian distribution if $v$ is concentrated on one
entry. These convergences require a finite fourth moment, and this paper
considers situations violating this condition. For symmetric distributions
$a_{11},$ heavy-tailed with index $\alpha \in (0,4),$ the fluctuations are
shown to be universal and dependent on $\theta$ but not on $v,$ whereas a
subfamily of the edge case $\alpha=4$ displays features of both the light- and
heavy-tailed regimes: two limiting laws emerge and depend on whether $v$ is
localized, each presenting a continuous phase transition at $\theta_0=1,
\theta_0 \in [1,\frac{128}{89}],$ respectively. These results build on our
previous which analyzes the asymptotic behavior of
$\lambda_1(\frac{1}{\sqrt{n}}A)$ in the aforementioned subfamily.
