Convergence of the imaginary parts of simplest fractions in $L_p(\mathbb R)$ for $p<1$

For $p\in(1/2,1)$, the $L_p(\mathbb R)$-convergence of the series $\sum_{k=1}^\infty|\operatorname{Im}(t-z_k)^{-1}|$ is studied, where the $z_k$ are some points on the complex plane. The problem is solved completely in the case where the sequence $\{\operatorname{Re}z_k\}$ has no limit points. Also, the case where this sequence has finitely many limit points is studied.