  We present Ir $4f$ core-level hard-x-ray photoemission spectroscopy (HAXPES)
experiments conducted across antiferromagnetic (AFM) ordering transition in
Ruddlesden-Popper iridates Sr$_2$IrO$_4$ and Sr$_3$Ir$_2$O$_7$. The Ir $4f$
spectra exhibit distinct changes between the AFM and paramagnetic (PM) phases,
with the spectral difference $I_\text{PM}-I_\text{AFM}$ showing a contrasting
behavior in the two compounds. By employing computational simulations using the
local-density approximation combined with the dynamical mean-field theory
method, we elucidate that $I_\text{PM}-I_\text{AFM}$ primary reflects the
Slater or Mott-Hubbard character of the AFM insulating state rather than
material specific details. This sensitivity to fine low-energy electronic
structure arises from the dependence of charge-transfer responses to the sudden
creation of a localized core hole on both metal-insulator transitions and
long-range AFM ordering. Our result broadens the applications of core-level
HAXPES as a tool for characterization of electronic structure.
