Well if you name the app someone might know whether this is normal for the app or not.
In general message notifications clear when you read the message - this works fine for email, SMS, even Facebook notifications (Facebook is my go-to example of a poorly-written app: it's quite happy to post a notification for a story that you read an hour ago, which is one reason I turn notifications off for the wretched thing). So if this app isn't doing that the best advice I can give at the moment is to contact the app developer, since it sounds like they are not handling notifications correctly (or at least not in the way you expect. There isn't a system setting for this, the app developer has chosen how they are cleared (or not), so unless there is a setting in the app it will be up to them to correct it.