I thought about this a bit on the context of some conversations I was having with Allen Knutson, where I wanted to be able to localize to generic points of Frobenius split subvarieties. I came to the following conclusions:
(1) The relationship between near splittings and sections of $\omega_X^{-p+1}$ goes through the Cartier operator, which only works over perfect fields. To be concrete, suppose that $a \in k$ is not a $p$-th power. Consider the form $a (dx)^{-p+1}$ on $\mathbb{A}^1$. What near splitting would you like to associate it to?
If you work over perfect fields I could not find any problems.
(2) As I mentioned, I wanted to be able to localize at generic points of subvarieties, which will not be perfect. In this case, I had the following idea: Let $S$ be a ring of characteristic $p$ equipped with a splitting $\phi: S \to S$. I think the correct thing to study is varieties over $S$ equipped with splittings which extend $\phi$. In this case, I think we can then pass to any localization of $S$, quotient by any split ideal of $S$ and do other things we'd like to do. If $S$ is a perfect field then it has a unique splitting, given by $p$-th root, so this includes the case of my previous bullet point.
I didn't write anything about this because I didn't find anything interesting to do with it. The answer may just be that no one has had a reason to write this up. Or, of course, there could be some major issue I missed.
