Why should have Peter May worked with CGWH instead of CGH in "The Geometry of Iterated Loop Space"? This is a follow-up to Dan Ramras' answer of this question.
The following correction can be found in the errata to The Geometry of Iterated Loop space (Page 484 here).

The weak Hausdorff rather than the Hausdorff property should be required of spaces [...] in order to validate some of the limit arguments used [...].

I was not able to figure out which type of "limit arguments" really would have needed the weak Hausdorff condition instead of the regular one used in the original paper and would be happy to understand the necessity of the correction, ideally by means of an explicit example in the paper.
 A: I'm not quite certain what Peter May had in mind 40 years ago,
but probably he had in mind the fact that pushouts are a lot better
behaved in CGWH than in CGH.  Specifically, CGWH is closed
under pushouts, one leg of which is the inclusion of a closed
subspace.  CGH does not have such nice behavior, and pushouts
like that are used all over The Geometry of Iterated Loop Spaces,
specifically in the construction of a monad from an operad and
in the use of geometric realizations of simplicial spaces.
A: I don't know for Peter May's work. I know that the weak Hausdorff condition can be useful for the following reason: let $f:X\to Y$ be a continuous map between Hausdorff k-spaces. Then consider the equivalence relation $\mathcal{R}_f$ on $X$ associated with $f$: $x \mathcal{R} y$ if and only if $f(x)=f(y)$. The graph of $\mathcal{R}_f$ is closed in $X\times X$ since it is the inverse image by $(f,f)$ of the diagonal of $Y$ which is closed in $Y\times Y$ because $Y$ is Hausdorff. The quotient set $X/\mathcal{R}_f$ equipped with the final topology is weak Hausdorff. There are examples (that I don't have in mind, maybe someone will be able to refresh my memory) where the quotient set $X/\mathcal{R}_f$ equipped with the final topology is not Hausdorff anymore. Of course, the "Hausdorffisation" of $X/\mathcal{R}_f$ still exists but the underlying set is not the quotient set anymore. More identifications can be done by the "Hausdorffisation" functor, i.e. we don't have the control anymore over what is identified in $X/\mathcal{R}_f$.
