By definition, a weakly Mahlo cardinal is a weakly inaccessible cardinal $\kappa$ such that $\{\alpha<\kappa : \alpha\text{ regular}\}$ is stationary in $\kappa$. The first requirement can be trivially simplified by asking that $\kappa$ be regular.
I'm interested in the case where $\kappa$ is singular of uncountable cofinality. That is,
Let $\mathrm{cf} \kappa >\omega$ and assume that $\{\alpha<\kappa : \alpha\text{ regular}\}$ is stationary in $\kappa$. Is it weakly Mahlo?
Actually, I'm mostly interested in the first such $\kappa$. I came across this while thinking of an example of a stationary $A\subseteq\kappa$ such that for all regular $\lambda<\kappa$, $A\cap \lambda$ is not stationary in $\lambda$.
Recall that if $\kappa$ is a limit cardinal, then the cardinals below $\kappa$ form a club; since the cofinality is uncountable, then the limit cardinals in fact form a club. This means that necessarily $\kappa$ has stationarily many inaccessible cardinals below.
If $\kappa$ is singular, then we can transfer this club to a club in its cofinality, say $\mu$. But now look at a club of order type $\mu$, its limit points all have cofinality less than $\mu$. In particular, no regular cardinals can be there, since regular cardinals with cofinality $<\mu$ are all $<\mu$.