  Limit theorems for the magnetization in the $p$-spin Curie-Weiss model, for
$p \geq 3$, has been derived recently by Mukherjee et al. (2021). In this
paper, we strengthen these results by proving Cram\'er-type moderate deviation
theorems and Berry-Esseen bounds for the magnetization (suitably centered and
scaled). In particular, we show that the rate of convergence is
$O(N^{-\frac{1}{2}})$ when the magnetization has asymptotically Gaussian
fluctuations, and it is $O(N^{-\frac{1}{4}})$ when the fluctuations are
non-Gaussian. As an application, we derive a Berry-Esseen bound for the maximum
pseudolikelihood estimate of the inverse temperature in $p$-spin Curie-Weiss
model with no external field, for all points in the parameter space where
consistent estimation is possible.
