The adverb hazily describes things done through a heavy mist or in a blurry way, like the view from your hotel room window of sailboats moving hazily past thick clouds.
You can use hazily for things that are figuratively foggy, too. You might only hazily remember your fifth birthday party — the memory is there, but it's clouded by time passing. Or you could find yourself only hazily aware that your sister is angry until she starts yelling at you. Hazily comes from the now-obsolete nautical term hawsey, "foggy."