  It has recently been discovered that a fraction of ultra-luminous X-ray
sources (ULXs) exhibit X-ray pulsations, and are therefore powered by
super-Eddington accretion onto magnetized neutron stars (NSs). For typical ULX
mass accretion rates ($\gtrsim 10^{19}\,{\rm g\,s^{-1}}$), the inner parts of
the accretion disc are expected to be in the supercritical regime, meaning that
some material is lost in a wind launched from the disc surface, while the rest
forms an optically thick envelope around the NS as it follows magnetic field
lines from the inner disc radius to the magnetic poles of the star. The
envelope hides the central object from a distant observer and defines key
observational properties of ULX pulsars: their energy spectrum, polarization,
and timing features. The optical thickness of the envelope is affected by the
mass losses from the disc. We calculate the mass loss rate due to the wind in
ULX pulsars, accounting for the NS magnetic field strength and advection
processes in the disc. We argue that the detection of strong outflows from ULX
pulsars can be considered evidence of a relatively weak dipole component of the
NS magnetic field. We estimate the influence of mass losses on the optical
thickness of the envelope and analyze how the envelope affects broadband
aperiodic variability in ULXs. We show that brightness fluctuations at high
Fourier frequencies can be strongly suppressed by multiple scatterings in the
envelope and that the strength of suppression is determined by the mass
accretion rate and geometrical size of the magnetosphere.
