Yes, Mighty does support personalized playlists such as the Daily Mixes? Most of your personalized playlists will be displayed in the Made For You category on the Browse screen. For any personalized playlist that is not displayed in the Made For You category, you can sync that playlist to Mighty by adding it to your Spotify library and accessing it through the root section of the Browse screen in the Mighty app.
Note that most personalized playlists are dynamically updated by Spotify on a set time interval. For example, the Daily Mixes are updated daily. Once synced to Mighty, these personalized playlists will not be automatically updated on Mighty. In order to update these playlists, connect Mighty to the app and WiFi, navigate to the Home screen, select the playlist, and tap Refresh. You can also enable Stay Fresh sync to have these playlists automatically updated each night. More information on Stay Fresh is available here.