  We investigate the possibility of studying the strange hidden-charm
pentaquark state $P_{cs}(4459)$ by photon-induced reactions on a proton target
in an effective Lagrangian approach. The production process is described by the
$t$-channel $K^{-}$ exchange, the $u$-channel $\Lambda$ exchange, the contract
term, and the $s$- channel nucleon pole. Our theoretical approach is based on
the assumption that $P_{cs}(4459)$ with $J^{P}=1/2^{-}$ or $J^{P}=3/2^{-}$ can
be interpreted as a molecule composed of $\bar{D}^{*}\Xi_c$. Using the coupling
constants of the $P^{J^P}_{cs}$ to $\gamma{}\Lambda$ and $K^{-}p$ channels
obtained from molecule picture of the $P^{J^{P}}_{cs}(4459)$, the total
cross-sections of the process $\gamma{}p\to{}P^{J^P}_{cs}K^{+}$ is evaluated.
Our calculation indicates that the cross-section for
$\gamma{}p\to{}P^{1/2^{-}}_{cs}K^{+}$ and $\gamma{}p\to{}P^{3/2^{-}}_{cs}K^{+}$
are of the order of 10.0 pb and 5.0 pb, respectively. In addition, we compute
the cross-section by assuming $P_{cs}(4459)$ as a compact pentaquark and find
it is quite different from the results of $\bar{D}^{*}\Xi_c$ molecule.
  Those results can be measured in future experiments, such as the Electron-Ion
Collider in China and the United States. And can be used to test the nature of
the $P_{cs}$.
