  A large fraction of galactic nuclei is expected to host supermassive black
hole binaries (BHB), likely formed during the early phase of galaxies assembly
and merging. In this paper, we use a large set of state-of-art numerical models
to investigate the interplay between a BHB and a massive star cluster (GCs)
driven toward the galactic centre by dynamical friction. Varying the BHB mass
and mass ratio and the GC orbit, we show that the reciprocal feedback exerted
between GCs and the BHB shapes their global properties. We show that, at
GC-to-BHB mass ratios above 0.1, the GC affects notably the BHB orbital
evolution, possibly boosting its coalescence. This effect is maximized if the
GC moves on a retrograde orbit, and for a non-equal mass BHB. We show that the
GC debris dispersed around the galactic nucleus can lead to the formation of a
nuclear cluster, depending on the BHB tidal field, and that the distribution of
compact remnants resulting from the GC disruption can carry information about
the BHB orbital properties. We find that red giant stars delivered by the
spiralling GC can be disrupted at a rate of $\simeq (0.7-7)\times 10^{-7}$
yr$^{-1}$ for BHB masses $\sim 10^7{\rm M}_\odot$, while tens to hundreds of
stars can be possibly observed in the galactic halo as high-velocity stars,
with velocities up to $\sim 2000$ km s$^{-1}$, depending on the BHB orbital
properties.
