  We construct Pleba\'nski-Demia\'nski stationary and axisymmetric solutions
with two expanding and double principal null directions in the framework of
Metric-Affine gauge theory of gravity. Starting from the new improved form of
the metric with vanishing cosmological constant recently achieved by Podolsk\'y
and Vr\'atn\'y, we extend this form in the presence of a cosmological constant
and derive the conditions under which the physical sources of the torsion and
nonmetricity tensors provide dynamical contributions preserving it in
Weyl-Cartan geometry. The resulting black hole configurations are characterised
by the mass, orbital angular momentum, acceleration, NUT parameter,
cosmological constant and electromagnetic charges of the Riemannian sector of
the theory, as well as by the spin and dilation charges of the torsion and
nonmetricity fields. The former is subject to a constraint representing a
decoupling limit with the parameters responsible of axial symmetry, beyond
which the geometry of the space-time is expected to be corrected.
