  We investigate the fluctuations of linear spectral statistics of a Wigner
matrix $W\_N$ deformed by a deterministic diagonal perturbation $D\_N$, around
a deterministic equivalent which can be expressed in terms of the free
convolution between a semicircular distribution and the empirical spectral
measure of $D\_N$. We obtain Gaussian fluctuations for test functions in
$\mathcal{C}\_c^7(\mathbb{R})$ ($\mathcal{C}\_c^2(\mathbb{R})$ for fluctuations
around the mean). Furthermore, we provide as a tool a general method inspired
from Shcherbina and Johansson to extend the convergence of the bias if there is
a bound on the bias of the trace of the resolvent of a random matrix. Finally,
we state and prove an asymptotic infinitesimal freeness result for independent
GUE matrices together with a family of deterministic matrices, generalizing the
main result from [Shl18].
