  In this work, the two-photon-exchange (TPE) effects in $ep\rightarrow
en\pi^+$ at small $-t$ are discussed within a hadronic model. Under the pion
dominance approximation the TPE contribution to the amplitude can be described
by a scalar function in the limit $m_e\rightarrow 0$. The TPE contributions to
the amplitude and the unpolarized differential cross section are both estimated
when only the elastic intermediate state is considered. We find that the TPE
corrections to the unpolarized differential cross section are about from $-4\%$
to $-20\%$ at $Q^2=1$--$ 1.6$ GeV$^2$. After considering the TPE corrections to
the experimental data sets of unpolarized differential cross section, we
analyze the TPE corrections to the separated cross sections
$\sigma_{\textrm{L,T,LT,TT}}$. We find that the TPE corrections (at
$Q^2=1$--$1.6$ GeV$^2$) to $\sigma_{\textrm{L}}$ are about from $-10\%$ to
$-30\%$, to $\sigma_{\textrm{T}}$ are about $20\%$, and to
$\sigma_{\textrm{LT,TT}}$ are much larger. By these analysis, we conclude that
the TPE contributions in $ep\rightarrow en\pi^+$ at small $-t$ are important to
extract the separated cross sections $\sigma_{\textrm{L,T,LT,TT}}$ and the
electromagnetic magnetic form factor of $\pi^+$ in the experimental analysis.
