The impact of the presence of the near-surface graded-gap layers with an increased content of CdTe on the admittance of MIS structures based on MBE-grown n-Hg1–xCdxTe (x = 0.22–0.23) with the Al2O3 insulating coating has been experimentally studied. It has been shown that the structures with a gradedgap layer are characterized by a deeper and wider capacitance dip in the low-frequency capacitance–voltage (CV) characteristic and by higher values of the differential resistance of the space-charge region than the structures without such a layer. It has been found that the main features of the hysteresis of capacitance dependences typical of the graded-gap structures with SiO2/Si3N4 are also characteristic of the MIS structures with the Al2O3 insulator. The factors that cause an increase in the CV characteristic hysteresis upon formation of the graded-gap layer in structures with SiO2/Si3N4 or Al2O3 are still debatable, although it may be assumed that oxygen plays a certain role in formation of this hysteresis