Yukawa potential in the Schrödinger equation The Coulomb potential:
$$
V_{C}(r)=-\frac{Z e^{2}}{4 \pi \epsilon_{0} r}
$$
Is the classical interaction energy between a particle with charge $-e$ and a potential with charge $Ze$.
Therefore, if I use this potential in the Schrödinger equation, I am describing, in a simple way, the electron of a hydrogen-like atom.
But, if I use the Yukawa potential:
$$
V_{Y}(r)=-\frac{g^{2}}{4 \pi} \frac{1}{r} e^{-\mu r}
$$
That is the strong interaction between hadrons (with $\mu$ the mass of the exchanged particle in the interaction),
Question: What system am I describing if I use this potential in the Schrödinger equation?
The term $e^{-\mu r}$ is the screening term.
Question: What is the physical meaning of adding this screening term to the Coulomb potential in the Schrödinger equation?
My intuition tells me that a screening term can be used to describe valence electrons of a multi-electron atom.
 A: 
What system am I describing if I use this potential in the Schrödinger equation?

If you use such a term (alone) you are damping the interaction in some sense. That means you are taking the Coulomb interaction which is generally long range (the potential decays as $1/r$ so that even far away points cannot always be assumed to be free) and turning it into a short range interaction (the exponential decay tells us that farther than a distance $1/\mu$ the potential is essentially zero). That means that a system which damps the Coulomb potential may be a candidate to be described by this type of potential. This is what people generally call screening. If you take an atom which is hydrogen-like, a first approximation is to just change the charge of the nucleus, however to higher precision, the cloud of electrons surrounding the nucleus might screen the nucleus' charge and modify the potential as you suggest.

What is the physical meaning of adding this screening term to the Coulomb potential in the Schrödinger equation?

I can recommend reading about this specific model Thomas-Fermi model and the Thomas-Fermi Screening
But as you can see both questions point towards the same answer. The Yukawa interaction you speak about is relevant in high-energy physics, under that name and when dealing with gauge bosons that are massive, leading to the exponential suppression. However at the atomic scale and the atoms you are speaking about, it can appear as the Wiki explains when modeling the possible effects of a a cloud of electrons with some smeared out charge density.
