Let $C$ be a projective smooth connected curve over an algebraically closed field $k$. Let $(P,G,p)$ be a triple, where $G$ is a finite $k$-group scheme, $P$ is a $G$-torsor over $C$, $p\in P(k)$ a rational point.
The question is: do we have a triple $(Q,H,q)$ with $Q$ a reduce scheme so that we have a morphism $(h,\rho):(Q,H,q)\to(P,G,p)$ where $\rho: H\to G$ is a group homomorphism and $h: Q\to P$ is a $C$-scheme morphism sending $q\mapsto p$ and interwines the action.
Remark1: The above condition is not always satisfied when $C$ is not smooth. There is a counterexample of a reduced projective connected curve. This counterexample is due to Deligne.
Remark2: For abelian varieties the above condition is always satisfied. In fact, Madhav V.Nori has proved in "The Fundamental Group-Scheme of an Abelian Variety" that for any triple $(P,G,p)$ over an abelian variety $A$ with $p$ lying over $0\in A(k)$, there is an isogeny $n: A\to A$ such that there is a map $(A, A[n], 0)\to (P,G,p)$ in the sense above.