The superconducting phase of the $mathrm{HgBa}_2mathrm{CuO}_{4+delta}$ (Hg-1201) and $mathrm{HgBa}_2mathrm{Ca}_2mathrm{Cu}_3mathrm{O}_{8+delta}$ (Hg-1223) cuprates has been investigated by Raman spectroscopy under hydrostatic pressure. Our analysis reveals that the increase of $T_c$ with pressure is slower in Hg-1223 cuprate compared to the Hg-1201 due to a charge carrier concentration imbalance (accentuated by pressure) between the $mathrm{CuO}_2$ layers of Hg-1223. We find that the energy variation under pressure of the apical oxygen mode from which the charge carriers are transferred to the $mathrm{CuO}_2$ layers, is the same for both the Hg-1223 and Hg-1223 cuprates and it is controlled by the inter-layer compressibility. At last, we show that the binding energy of the Cooper pairs related to the maximum amplitude of the $d-$ wave superconducting gap at the anti-nodes, does not follow $T_c$ with pressure. It decreases while $T_c$ increases. In the particular case of Hg-1201, the binding energy collapses from 10 to 2 $K_B T_c$ as the pressure increases up to 10 GPa. These direct spectroscopic observations joined to the fact that the binding energy of the Cooper pairs at the anti-nodes does not follow $T_c$ either with doping, raises the question of its link with the pseudogap energy scale which follows the same trend with doping.
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