Are the Missions available only on the Desktop App?
At this particular moment, yes.
The Musixmatch team is working hard to bring the Missions to mobile in the smartest way, developing the best experience that Curators could ever imagine.
For now, though, you can only complete the Missions on the Desktop App (always make sure to have the newest version downloaded) on Windows 10 and macOS.
Download the Desktop App now, your Mission is waiting for you!