  The axino, the supersymmetric partner of axion, is a well-motivated warm/hot
dark matter candidate, and provides a natural solution to the relic density
problem for the bino-like neutralino if it is the lightest supersymmetric
particle (LSP). With the Generalized Minimal Supergravity, we study such kind
of the viable parameter space where the bino-like neutralino is the next-to-LSP
(NLSP) and the axino is the LSP. In addition, we consider a scenario where the
bino is a long-lived NLSP with the lifetime varying from $10^{-6}$s to
$10^{-4}$s, and then propose a new signal searching scheme involving one
displaced photon together with the large missing transverse momentum at the
HL-LHC. The bino-like lightest neutralino lies under or around 100 GeV and is
produced as a decay product of the right-handed sleptons.The relevant axion
coupling $f_a$ can be probed up to $\mathcal{O}(10^9)$ GeV at 2$\sigma$ level
for the right-handed slepton mass under 300 GeV and the lightest neutralino
mass under 100 GeV.
