  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 Tc with
pressure is slower in Hg-1223 cuprate compared to the Hg-1201 due to a charge
carrier concentration imbalance accentuated by pressure in Hg-1223. We find
that the energy variation under pressure of the apical oxygen mode from which
the charge carriers are transferred to the $\mathrm{CuO}_2$ layer, 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 Tc with pressure. It decreases while Tc
increases. In the particular case of Hg-1201, the binding energy collapses from
10 to 2K$_B$Tc 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 Tc either with doping, raises
the question of its link with the pseudogap energy scale which follows the same
trend with doping.
