Sr$_2$IrO$_4$ is the archetype of the spin-orbit Mott insulator, but the nature of the metallic states that may emerge from this type of insulator is still not very well known. We study with angle-resolved photoemission the insulator-to-metal transition observed in Sr$_2$Ir$_{1-x}$Rh$_x$O$_4$ when Ir is substituted by Rh (0.02 < $x$ < 0.35). The originality of the Rh doping is that Ir and Rh, which are formally isovalent, adopt different charge states, a rather unusual and inhomogeneous situation. We show that the evolution to the metallic state can be essentially understood as a shift of the Fermi level into the lower Hubbard band of Sr$_2$IrO$_4$. The Mott gap appears quite insensitive to the introduction of up to $sim$20% holes in this band. The metallic phase, which forms for $x$ > 0.07, is not a Fermi liquid. It is characterized by the absence of quasiparticles, unrenormalized band dispersion compared to calculations and an $sim$30-meV pseudo-gap on the entire Fermi surface.
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