  We consider the $pp \to pp \rho^{0} \pi^{0}$ and $pp \to pn \rho^{0} \pi^{+}$
processes at LHC energies. Our description is based on the nonperturbative
framework of tensor pomeron and tensor reggeon exchanges. We discuss the
Drell-Hiida-Deck type mechanism with centrally produced $\rho^{0}$ meson
associated with a very forward/backward $\pi N$ system. The considered
processes constitute an inelastic (non-exclusive) background to the $p p \to p
p \rho^0$ reaction in the case when only the centrally produced $\rho^{0}$
meson decaying into $\pi^{+} \pi^{-}$ is measured, the final state protons are
not observed, and only rapidity-gap conditions are checked experimentally. We
compare our results for the $\gamma \pi^{+} \to \rho^{0} \pi^{+}$ reaction with
the experimental data obtained by the H1 collaboration at HERA. We present
several differential distributions for the $pp \to pn \rho^{0} \pi^{+}$
reaction and estimate the size of the proton dissociative background to the
exclusive $p p \to p p \rho^0$ process. The ratio of integrated cross sections
for the inelastic $p p \to p N \rho^0 \pi$ processes, where $p N \rho^0 \pi$
stands for $p n \rho^0 \pi^{+}$ plus $p p \rho^0 \pi^{0}$, to the reference
reaction $pp \to \ pp \rho^0$ is of order of (7--10)\%. We present also the
ratios of the $\rho^0$ rapidity and transverse momentum distributions for the
inelastic $p p \to p N \rho^0 \pi$ versus the elastic $pp \to \ pp \rho^0$
reaction. Our results may be used to investigate the $\gamma \pi \to \rho^{0}
\pi$ process at LHC energies.
