Let $X$ be a variety over a field $F$ of characteristic $p>0$.
The usual (not 100% sure) definition of weak normality should be
$X$ is weakly normal if whenever $f:Y\to X$ is a morphism of varieties satisfy
1) $f$ is a finite birational bijective morphism, and
2) for every $p\in > X,q\in Y$, $f(q)=p$, the field extension $k(p)\to k(q)$ is purely inseparable.
Then, $f$ is an isomorphism.
In Brion and Kumar's book Frobenius splitting methods in Geometry and Representation Theory. The definition (1.2.3) of weak normality in the book is different, the condition 2) is dropped. Also, in the proof of the proposition (1.2.5) that every Frobenius split variety is weakly normal, I think the authors somehow use condition 2) in their proof.
I would want to know if condition 2) is redundant in the definition of weak normality, and I am happy to see a reference for a counter example of proposition (1.2.5) if we do not impose condition 2) in the definition. Thank you in advance.