That sounds more like an issue with the overall design rather than a mobile specific issue. It's less relevant for a game like this that has literally no chance of ever winding up on a console, but mouseover popups are a design element that only works on computers, and are very much something you put a fallback for when designing a game. Even if that fallback is as simple as a minimum display time for the popup. If working with something like that on my own game, I'd actually do a minimum display time in combination with suppressing the current popup if a new one is displayed. (In other words, if you're moving through the page only the latest popup will show, and the latest popup has a minimum amount of time before it goes away)
Of course, this works significantly better with a control setup that allows for multiple styles of input. A menu compatible with keyboard/controller will be compatible with popup/selection display by default, and allowing a popup to show on press and hold (rather than that qualifying as a click) could address some issues with touchscreens. Basically, what I'm saying is that there are tons of ways to handle it that are generally cross compatible.