Many of the lever activated drains use a tub that gets lifted by the lever. When it is down, it closes off the actual drain hole. When it is up, it is above the drain hole. Sometimes the adjustment gets loose, and it never uncovers the hole completely. Sometimes, gunk gets caught in it. In either case, you can usually take off the cover and pull it out. If you then clean things off, you can probably put it back and all will be well again, ready to use if you want to. It may be that the stopper has nothing to do with the slow tub draining. By pulling this out, you can access the drain with a snake and clean it out. That probably would fix things.