  The dynamical system studied in previous papers of this series, namely a
bound time-like geodesic motion in the exact static and axially symmetric
space-time of an (originally) Schwarzschild black hole surrounded by a thin
disc or ring, is considered to test whether the often employed
"pseudo-Newtonian" approach (resorting to Newtonian dynamics in gravitational
potentials modified to mimic the black-hole field) can reproduce phase-space
properties observed in the relativistic treatment. By plotting Poincar\'e
surfaces of section and using two recurrence methods for similar situations as
in the relativistic case, we find similar tendencies in the evolution of the
phase portrait with parameters (mainly with mass of the disc/ring and with
energy of the orbiters), namely those characteristic to weakly non-integrable
systems. More specifically, this is true for the Paczy\'nski--Wiita and a newly
suggested logarithmic potential, whereas the Nowak--Wagoner potential leads to
a different picture. The potentials and the exact relativistic system clearly
differ in delimitation of the phase-space domain accessible to a given set of
particles, though this mainly affects the chaotic sea whereas not so much the
occurrence and succession of discrete dynamical features (resonances). In the
pseudo-Newtonian systems, the particular dynamical features generally occur for
slightly smaller values of the perturbation parameters than in the relativistic
system, so one may say that the pseudo-Newtonian systems are slightly more
prone to instability. We also add remarks on numerics (a different code is used
than in previous papers), on the resemblance of dependence of the dynamics on
perturbing mass and on orbital energy, on the difference between the Newtonian
and relativistic Bach--Weyl rings, and on the relation between Poincar\'e
sections and orbital shapes within the meridional plane.
