Approximation of subharmonic functions of slow growth

Let $u$ be a subharmonic function in $\mathbb C$, $\mu_u$ its Riesz measure. Suppose that $C_1\le\mu(\{z:R<|z|\le R\psi(R)\}\le C_2$ $(R\ge R_1)$ for some positive constants $C_1$, $C_2$, and $R_1$, and a slowly growing to $+\infty$ function $\psi(r)$ such that $r/\psi(r) \nearrow +\infty$ ($r\to+\infty$). Then there exist an entire function $f$, constants $K_1=K_1(C_1,C_2)$, $K_2=K_2(C_2)$ and a set $E\subset\mathbb C$ such that
$$ |u(z)-\log|f(z)||\le K_1\log\psi(|z|), \qquad z\to\infty, \quad z\notin E, $$
and $E$ can be covered by the system of discs $D_{z_k}(\rho_k)$ satisfying
$$ \sum_{R<|z_k|<R\psi(R)}\frac{\rho_k\psi(|z_k|)}{|z_k|}<K_2, $$
as $R_2\to+\infty$. We prove also that the estimate of the exceptional set is sharp up to a constant factor.