Both absolutely get me every time. The weather is always present, say you hope the "bad weather stays away", and everyone has mental health, people struggle with "mental health illness/issue" or "poor mental health"
Have to disagree also, on the basis that everyone knows quite clearly what you're talking about if you say this. Plus if you said that someone was struggling with their health, it would be plainly obvious it meant 'bad health' otherwise they wouldn't be struggling with it. It's just a bit of colour in the language if anything, as opposed to being exactly technically correct.