I’m interested in the zeroes of the complex function

$f(z,\bar{z}) = p(z) + \frac{1}{log(|z|)} q(z)$

where both $p$ and $q$ are polynomials of the complex variable $z$ (and are therefore holomorphic). I’m interested even in restricted cases, such as when both $p$ and $q$ are of low degree. I would also be very happy if anything could be said about the case where $p$ and $q$ are Kac-polynomials of high degree.

While I don’t know the literature well, I am aware that there are beautiful results relating to the clustering of zeroes on the unit circle in the context of random polynomials --- the simplest being Kac-polynomials. Presumably the factor of $\frac{1}{log(|z|)}$ spoils the usage of these results though.

I am also vaguely aware of Rouche’s theorem. However, since $f$ is not holomorphic, this result can not be used directly.

I would like to know how to count the zeroes of this function. Treating the coefficients as independent and identically distributed random variables would be fine too if that would be useful. To be honest I have no idea where to even start looking, so any and all suggestions are most welcome.