  While recent experiments have found that at optimum doping for the highest
critical temperature in HgBa$_2$CuO$_{4+y}$ (Hg1201) the oxygen interstitials
(O-i) are not homogeneously distributed but form one-dimensional atomic wires,
there are no available information of its electronic structure considering
self-organized O-i atomic wires. Here we report the calculated electronic
structure of HgBa$_2$CuO$_{4+y}$ where oxygen interstitials form atomic wires
along (1,0,0) crystal direction in the Hg layer. We find that at optimum doping
for superconductivity the chemical potential is tuned near an electronic
topological Lifshitz transition for the appearing of a second quasi 1D Fermi
surface. A $first$ large Fermi surface coexists with a $second$ incipient quasi
one dimensional (1D) Fermi surface related with atomic wires of oxygen
interstitials. Increasing oxygen doping the chemical potential is driven to the
band edge of the $second$ 1D-band giving a peak in the density-of-states. The
new 1D electronic states are confined near the oxygen interstitial wires with a
small spread only on nearby sites. Spin-polarized calculations show that the
magnetic response is confined in the oxygen-poor domains free of oxygen
interstitials wires and it is quite insensitive to the density of O-i wires.
