Here is perhaps the simplest example of a CGWH space which fails to be Hausdorff.

Start with a countable metric space $X$ so that with one exception $x$, each point is open, but so that at the exceptional point, $X$ is not locally compact at $x$. 

It is easy to find such a subspace of the real line. 
(Start with $0$ and$ (1/n)+(1/(m+n)$). Now delete each $1/n$).

Let $Y$ be the one point compactification, adding to $X$ a new point $y$, whose neighborhood complements are compact in $X$. In the new space $Y$, compact subsets are closed (and in particular $Y$ is WH), but $x$ and $y$ are inseparable.

See for example, Example 99 from [Counterexamples in Topology](https://en.wikipedia.org/wiki/Counterexamples_in_Topology) by Steen and Seebach.