Does red shift need to compensate for space expansion
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.