  This paper extends a previous bound of order $O(n^{-1})$ of the authors
(arXiv:1405.7820[math.PR]), for the rate of convergence in Kolmogorov distance
of the expected spectral distribution of a Wigner random matrix ensemble to the
semicircular law. Here we relax the moment conditions for entries of the Wigner
matrices from order $8$ to order $4+ \epsilon$ for an arbitrary small
$\epsilon>0$.

Probability