  In this paper, a simplified second-order Gaussian Poincar\'e inequality for
normal approximation of functionals over infinitely many Rademacher random
variables is derived. It is based on a new bound for the Kolmogorov distance
between a general Rademacher functional and a Gaussian random variable, which
is established by means of the discrete Malliavin-Stein method and is of
independent interest. As an application, the number of vertices with prescribed
degree and the subgraph counting statistic in the Erd\"os-R\'enyi random graph
are discussed. The number of vertices of fixed degree is also studied for
percolation on the Hamming hypercube. Moreover, the number of isolated faces in
the Linial-Meshulam-Wallach random $\kappa$-complex and infinite weighted
2-runs are treated.
