The self-screening error is the part of the self-interaction error which causes each electron to artificially screen its own presence.
It remains within the $GW$ approximation if the exact dynamically screened Coulomb interaction $W$ is used.
Let's consider adding an electron to a system which did not have any electrons before. The energy associated is simply:
$$E_{N+1=1}-E_{N=0}=\epsilon_1$$
or just the energy of that electron at that particular level.
DFT, HF, and GW level of theories return the addition energy as expected (i.e. the energy of the level in their corresponding level of theories).
$$\left( -\frac{\nabla^2}{2}+V_0(x_1)\right)\phi_1(x_1)=\epsilon_1 \phi_1 (x_1)$$
Similarly removing the last electron from the system gives the removal energy.
DFT and HF again fulfills the "correct" expectation, returning the corresponding energy level,
$$\left( -\frac{\nabla^2}{2}+V_0(x_1)\right)\phi_1(x_1)=\epsilon_1 \phi_1 (x_1)$$
However, in COHSEX approximation, GW returns
$$\left( -\frac{\nabla^2}{2}+V_0(x_1)+v_H(x_1)right)\phi_1(x_1)-\int dx_2|left(\phi_1(x_1)\phi_1^*(x_2)W(x_1,x_2)+\delta(x_1-x_2)\frac{1}{2}W_p(x_1,x_2)\right)\phi_1(x_1)=\epsilon_1^\text{GW}\phi_1(x_1)$$
where $W_p=W-v$
The problem is the screening term. The extracted particle screens itself! Hence
$$\epsilon_1^\text{GW}\neq\epsilon_1$$
This is a bad treatment of the induced exchange.
This error can be correct by using an approximate vertex term like
$$\Gamma=\begin{cases} \delta+f_{xc}P & \text{valance} \\ \delta & \text{conduction} \end{cases}$$
where $f_{xc}$ is built from TDDFT with the total term, and contains the quasiparticle contribution.
This approximate term does not correct the atomic limit problem, which is another discussion.