The software here is focused on the forums, and the PMs are an afterthought as we want to encourage people to post to the forums and create a permaculture resource for future readers.
That said, there should be a 'delete this message' button when you open the Purple Moosage. On my screen, it's on the bottom right. Be advised this button only removes the moosage from your view and doesn't remove the moosage from the system or the other persons moosage box. I don't know if this is visible in the mobile view.