We study the distribution of zeroes of power series with infinite radius of convergence. The coefficients of the series have the form $ξ(n)a(n)$, where $a$ is a smooth sequence of positive numbers, and $ξ$ is a sequence of complex-valued multipliers having binary correlations and no gaps in the spectrum. We show that under certain assumptions on the smoothness of the sequence $a$ and on the binary correlations of the multipliers $ξ$, the zeroes of the power series are equidistributed with respect to a radial measure defined by the sequence $a$. We apply our approach to several examples of the sequence $ξ$: (i) IID sequences, (ii) sequences $e(αn^2)$ with Diophantine $α$, (iii) random multiplicative sequences, (iv) the Golay--Rudin--Shapiro sequence, (v) the indicator function of the square-free integers, (vi) the Thue--Morse sequence.