  The superconducting phase of the $\mathrm{HgBa}_2\mathrm{CuO}_{4+\delta}$
(Hg-1201) and $\mathrm{HgBa}_2\mathrm{Ca}_2\mathrm{Cu}_3\mathrm{O}_{8+\delta}$
(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.
