  Let $\calO_K$ be a complete discrete valuation ring of mixed characteristic
$(0,p)$ with a perfect residue field. In this paper, for a semi-stable $p$-adic
formal scheme $\frakX$ over $\calO_K$ with rigid generic fibre $X$ and
canonical log structure $\calM_{\frakX} = \calO_{\frakX}\cap\calO_X^{\times}$,
we study Hodge--Tate crystals over the absolute logarithmic prismatic site
$(\frakX,\calM_{\frakX})_{\Prism}$. As an application, we give an equivalence
between the category of rational Hodge--Tate crystals on the absolute
logarithmic prismatic site $(\frakX,\calM_{\frakX})_{\Prism}$ and the category
of enhanced log Higgs bundles over $\frakX$, which leads to an inverse Simpson
functor from the latter to the category of generalised representations on
$X_{\proet}$.
