The Concentration Function of Additive Functions with Special Weight

Suppose that $g(n)$ is an additive real-valued function,
$$W(N)=4+\min_\lambda\lambda^2+\sum_{p<N}\frac1p\min(1,(g(p)-\lambda\log p)^2), \quad E(N)=4+\sum_{p<N,\ g(p)\ne0}\frac1p.$$
In this paper, we prove the existence of constants $C_1$, $C_2$ such that the following inequalities hold:
$$\begin{aligned} &\sup_a|\{n,m,k:m,k\in\mathbb Z,\ n\in\mathbb N,\ n+m^2+k^2=N,\ g(n)\in[a,a+1)\}| \le\frac{C_1N}{\sqrt{W(N)}}, \\ &\sup_a|\{n,m,k:m,k\in\mathbb Z,\ n\in\mathbb N,\ n+m^2+k^2=N,\ g(n)=a\}| \le\frac{C_2N}{\sqrt{E(N)}}. \end{aligned}$$
The obtained estimates are order-sharp.