Given a sequence of real numbers $c_k\to-\infty$, is there always a $C^\infty$ subharmonic function $f$ on $\mathbb R^2$ and a sequence $z_k\to\infty$ with $|z_k|<k$ such that $$\displaystyle\limsup_{z\to\infty} \frac{f(z)}{\log |z|}<\infty\ \ \text{and}\ \ f(z_k)<c_k\ ?$$

I do not even know the answer for non-smooth $f$. In Hayman's "Subharmonic functions", volume 2, in example 7.26 on page 449 there is a weaker result that there is subharmonic $f$ of sub-logarithmic growth (in the sense above) and $z_k\to\infty$ such that $f(z_k)<-\log|z_k|$.

The question came up in research but it seems like an interesting question by itself.