  We investigate the role of electron correlation and disorder on the
electronic structure of layered nickelate (La$_{0.5}$Sr$_{0.5}$)$_2$NiO$_4$
using core level and valence band photoemission spectroscopy in conjunction
with density functional theory (DFT) and dynamical mean field theory (DMFT)
calculations. Sr 3$d$ and La 4$d$ core level spectra exhibit multiple features
associated with photoemission final state effects. An increase of unscreened
features in the Sr 3$d$ and La 4$d$ core level spectra with lowering
temperature suggests the reduction in density of states (DOS) at the Fermi
level, $E_F$. Valence band spectra collected using different photon energies
reveal finite intensity at $E_F$ and overall spectra are well captured by
DFT+DMFT. Strong renormalization of partially filled $e_g$ bands in DFT+DMFT
result indicates strong correlation in this system. Mass enhancement factor,
$m^*/m_{\text{DFT}} \sim$ 3, agrees well with values obtained from specific
heat measurements. High resolution spectra in the vicinity of $E_F$ show
monotonically decreasing spectral intensity with lowering temperature, which
evolves to exhibit a Fermi cut-off at low temperatures indicating metallic
character in contrast to insulating transport, suggesting Anderson insulating
state. $|E-E_F|^{1/2}$ dependence of the spectral DOS and square root
temperature dependence of spectral DOS at $E_F$ evidences the role of disorder
in the electronic structure of (La$_{0.5}$Sr$_{0.5}$)$_2$NiO$_4$.
