Does red shift need to compensate for space expansion Distant galaxies are moving as space expands, not moving through space.  So a photon coming to us from them has to come back through distorted space.  So is not spacetime distorted, so the photon has to go through an area where time slows down.  So would this cause red shift.  So maybe we have a simplistic view of red shift and maybe those galaxies are not going as fast as we think
Redshift is the "stretching" of the wavelength of light as it travels, over finite time, towards us, due to the ongoing expansion of space.

The effects of this redshift are indistinguishable from that caused by the doppler effect, so that is why some elementary treatments choose to discuss it in that way.

Most galaxies only have peculiar velocities with respect to thei comoving volume of a few hundred km/s, and this does perturb the cosmological redshift by that (small) amount. That is why Hubble's law only applies at distances such that cosmological expansion dominates.