In this paper, the fluctuations of linear spectral statistics of large random covariance matrices are shown to be gaussian, in the regime where both dimensions of the matrices go to infinity at the same pace. The main improvements with respect to Bai and Silverstein's CLT (2004) are twofold. First, we consider general entries with finite fourth moment, but whose fourth cumulant is non-null, i.e. whose fourth moment may differ from the moment of a (real or complex) Gaussian random variable. As a consequence, extra terms proportional to the (square of) the second non-absolute moment and to the fourth cumulant appear in the limiting variance and in the limiting bias, which not only depend on the spectrum of the population covariance matrix but also on its eigenvectors. Second, we relax the analyticity assumption over the underlying function by representing the linear statistics with the help of Helffer-Sjöstrand's formula. The CLT is expressed in terms of vanishing Lévy-Prohorov distance between the linear statistics' distribution and a Gaussian probability distribution, the mean and the variance of which depend upon the dimensions of the matrices and may not converge.