I was pinged earlier in a chat room with an abusive message. Because I wasn't around at the time it went right to my inbox.
Chat moderation removed the message before I even knew it existed, which I'm very thankful for. But the notification about that message is still in my inbox, so I still have a pretty vile message directed at me sitting there in my global inbox. I don't think that should exist by design; it's a way to ensure your abuse gets across to a person even if it's later removed.
So my suggestion: If a message is deleted as rude/abusive in chat, any inbox notifications coming from its mentions should be deleted too.
I would also accept just being able to delete items from my inbox outright. I think this is better though—if someone's getting abused in chat we shouldn't have to make them wade through cleaning up the mess themselves, we should ensure they don't have to see the abuse to begin with.
I'm going to get some folks to ping me to clear it out, so it'll be gone, but that's a workaround, and it's work I'd like others dealing with abuse to not have to go through.