Prismatic crystals for smooth schemes in characteristic $p$ with Frobenius lifting mod $p^2$

Let $(A,(p))$ be a crystalline prism with $A_n=A\mathrm{/}{p}^{n+1}A$ for all $n\ge 0$. Let ${\text{frakX}}_0$ be a smooth scheme over $A_0$. Suppose that ${\text{frakX}}_0$ admits a lifting ${\text{frakX}}_n$ over $A_n$ and the absolute Frobenius ${\text{rF}}_{{\text{frakX}}_0}:{\text{frakX}}_0\to {\text{frakX}}_0$ admits a lifting over $A_1$. Then we show that there is an equivalence between the category of the prismatic crystals of truncation $n$ on $({\text{frakX}}_0\mathrm{/}A{)}_{\text{Prism}}$ and the category of $p$-connections over ${\text{frakX}}_n$, which is compatible with cohomologies. This generalises a previous work of Ogus. We also give some remarks on trivializing the Hodge–Tate gerbe ${\pi }_{{\text{frakX}}_0}^{\mathrm{H}\mathrm{T}}:{\text{frakX}}_0^{\mathrm{H}\mathrm{T}}\to {\text{frakX}}_0$ introduced by Bhatt–Lurie.