  Fix a smooth projective family of curves $C \to S$ and a split reductive
group scheme $G$ over a Noetherian base scheme $S$. For any (possibly
nonreduced) fixed relative Cartier divisor $D$, we provide a treatment of the
moduli of $G$-bundles on the fibers of $C$ equipped with $t$-connections with
pole orders bounded by $D$. Under mild assumptions on the characteristics of
all the residue fields of $S$, we construct a Hodge moduli space $M_{Hod, G}
\to \mathbb{A}^1_S$ for the semistable locus, construct a Harder-Narasimhan
stratification, and thus obtain a semistable reduction theorem. If all the
fibers of the divisor of poles $D$ are nonempty, then we show that the stack of
semistable objects is smooth over $\mathbb{A}^1_{S}$. We also define a
Hodge-Hitchin morphism in positive characteristic and prove that it is proper.
