If the music you're playing via the music-playing app stops when you reopen Super Mario Run, try the following:
- Swipe down on the notification area to display the notification drawer.
- Press the music app's Play button in the notification drawer.
Note: On most Android devices, swiping down from the top of the screen while Super Mario Run is open will display the notification drawer. If this doesn't work, please contact the maker of your device.