  The model under consideration is the classical two-dimensional one-component
plasma (jellium) of pointlike particles with charge $e$, interacting pairwisely
via the logarithmic Coulomb potential and immersed in a uniform neutralizing
background charge density. The system is in thermal equilibrium at the inverse
temperature $\beta$, its thermodynamics depends only on the coupling constant
$\Gamma=\beta e^2$. We put into an infinite (homogeneous and translationally
invariant) plasma a guest particle of charge $Ze$ with $Z>-2/\Gamma$ in order
to prevent from the collapse of the jellium charges onto it. The guest particle
induces a screening cloud (the excess charge density) in the plasma. The zeroth
and second moments of this screening cloud were derived previously for any
fluid value of $\Gamma$. In this paper, we propose a formula for the fourth
moment of the screening cloud. The derivation is based on the assumption that
the fourth moment is, similarly as the second moment, analytic in $Z$ around
$Z=0$. An exact treatment of the limit $Z\to\infty$ shows that it is a finite
(cube) polynomial in $Z$. The $\Gamma$-dependence of the polynomial
coefficients is determined uniquely by considering the limits $Z\to 0$ and
$Z\to\infty$, and the compressibility sum rule for $Z=1$. The formula for the
fourth moment of screening cloud is checked in the leading and first correction
orders of the Debye-H\"uckel limit $\Gamma\to 0$ and at the exactly solvable
free-fermion point $\Gamma=2$. Sufficient conditions for sign oscillations of
the induced charge density which follow from the second-moment and
fourth-moment sum rules are discussed.
