Assertion is correct, 

But the work function only depends on the photoelectric metal.

Depth Dependence of Work Function in Metals
Metal surfaces exhibit a phenomenon where the work function, the minimum amount of energy required to remove an electron from the surface, varies as a function of depth. This depth dependence of the work function has important implications for the photoelectric effect.

Photoelectric Effect and Kinetic Energy Spread
When monochromatic light is incident on a metal surface, photoelectrons are emitted. These photoelectrons have a spread of kinetic energies, which can be attributed to the varying work function of the metal at different depths.

Explanation
The work function of a metal is related to the energy required to remove an electron from the surface. As we move deeper into the metal, the atomic environment changes, leading to differences in the strength of the electron's binding to the surface. This variation in binding energy results in a depth-dependent work function.
When light interacts with the metal surface, electrons are emitted from different depths within the material. Electrons originating from deeper layers experience a higher work function compared to those from the surface layers. As a result, the emitted electrons exhibit a range of kinetic energies, reflecting the varying work functions at different depths.
This depth dependence of the work function contributes to the spread of kinetic energies observed in photoelectrons produced by monochromatic light incident on a metal surface. Understanding this phenomenon is crucial for accurately interpreting experimental results in the field of surface science and photoemission spectroscopy.