  We present a theoretical investigation of the effects of correlations on the
electronic structure of the Mott insulator Sr$_2$IrO$_4$ upon electron doping.
A rapid collapse of the Mott gap upon doping is found, and the electronic
structure displays a strong momentum-space differentiation at low doping level:
The Fermi surface consists of pockets centered around $(\pi/2,\pi/2)$, while a
pseudogap opens near $(\pi,0)$. Its physical origin is shown to be related to
short-range spin correlations. The pseudogap closes upon increasing doping, but
a differentiated regime characterized by a modulation of the spectral intensity
along the Fermi surface persists to higher doping levels. These results,
obtained within the cellular dynamical mean-field theory framework, are
discussed in comparison to recent photoemission experiments and an overall good
agreement is found.
