Poly(N-isopropylacrylamide)-based microgels are soft colloids undergoing a volume phase transition (VPT) close to ambient temperature. Although widely employed for fundamental research and application purposes, the modifications of the microgel internal structure occurring at the VPT are not yet completely understood, especially concerning the role of electrostatics. Here we study in detail, both experimentally and numerically, the effect of the addition of acrylic acid (AAc) comonomer on the microgel deswelling process. By combining viscosimetry, light scattering, and electrophoresis, we show that the progressive addition of AAc increases the microgel size, modifying their mass-to-permeability ratio, and suppresses the occurrence of the VPT, progressively shifting the microgel collapse to higher temperatures. Most interestingly, however, we find that AAc addition highly enhances the two-step deswelling of these submicronsized networks, so that the inner core collapses at temperatures always lower than those marking the transition of the outer corona. These results indicate that a net increase of the charge-density mismatch between the bulk and the surface of the microgels takes place. Numerical simulations fully confirm this scenario and clarify the impact of the charge distribution on the two-step deswelling, with mobile counterions efficiently screening the charges within the inner core while leaving more monomers ionized on the surface. Our work unambiguously shows how electrostatic interactions influence the behavior of thermosensitive microgels in an aqueous environment close to the VPT.