  Given a domain $\Omega \subseteq \mathbb{R}^2$, let $\mathcal{D}(\Omega,x,R)$
be the number of lattice points from $\mathbb{Z}^2$ in $R\Omega-x$, for $R \ge
1$ and $x\in \mathbb{T}^2$, minus the area of $R\Omega$:
$$\mathcal{D}(\Omega,x,R) = \# \{ (j,k) \in \mathbb{Z}^2 :(j-x_1,k-x_2) \in
R\Omega \} - R^2|\Omega|.$$ We call
$\int_{\mathbb{T}^2}|\mathcal{D}(\Omega,x,R)|^pdx$ the $p$-th moment of the
discrepancy function $\mathcal{D}$. In 2014, Huxley showed that for convex
domains with sufficiently smooth boundary, the fourth moment of $\mathcal{D}$
is bounded by $\mathcal{O}(R^2\log R)$, and in 2019, Colzani, Gariboldi and
Gigante extended this result to higher dimensions. In this paper, our
contribution is twofold: first, we present a simple direct proof of Huxley's
2014 result; second, we establish new estimates for the $p$-th moments of
lattice point discrepancy of annuli of radius $R$, and any fixed thickness
$0<t<1$ for $p\ge 2.$
