  Ly$\alpha$ emission nebulae are ubiquitous around high-z galaxies and are
tracers of the gaseous environment on scales out to >100 kpc. High-z radio
galaxies (HzRGs, type-2 radio-loud quasars) host large scale nebulae observed
in the ionised gas differ from those seen in other types of high-z quasars. In
this work, we exploit MUSE observations of Lya nebulae around eight HzRGs
($2.9<z<4.5$). All the HzRGs have large scale Lya emission nebulae with seven
of them extended over 100 kpc at the observed surface brightness (SB) limit.
Because the emission line profiles are significantly affected by neutral
hydrogen absorbers across the entire nebulae, we perform an absorption
correction to infer maps of the intrinsic Lya SB, central velocity and velocity
width, all at the last scattering surface of the observed Lya photons. We find:
(i) The intrinsic SB radial profiles can be described by an inner exponential
and an outter power law; (ii) our HzRGs have higher SB and more asymmetric
nebulae than both RL and RQ type-1s; (iii) intrinsic nebula kinematics of four
HzRGs show evidence of jet-driven outflows but no general trends for the whole
sample; (iv) a relation between the nebula maximum extent and the offset
between the AGN and the nebula centroids; (v) an alignment between radio jet
position angles and the nebula morphology. All support a scenario where the
orientation of the AGN has an impact on the observed nebular morphologies and
resonant scattering may affect the shape of the SB profiles, nebular kinematics
and relations between the observed Lya morphologies. Furthermore, we find
evidence showing that the outskirts of the ionised gas nebulae may be
'contaminated' by Lya photons from nearby emission halos. Overall, this work
provides results which allow us to compare Lya nebulae around various classes
of quasars at and beyond Cosmic Noon. [Abridged]
