Prožura is a beautiful Bay on the NE side of Mljet Island. I even found a roadmap of Croatia with the cover photo of this bay on it.
Prožura is quite safe for anchoring except for the depth, which is too deep. You can tie your boat to a buoy of one of two restaurants or anchor by one of two smaller islands Planjak or Snjevci. The second restaurant has no buoys. You can anchor in the inner most part of the bay at about 4-10 meters.
Along Planjak the depth quickly deepens to 15 and more meters so you have to be close by. Snjevci is better but there is room enough for only two or three boats. Seabed there is sand and anchor holds well at about 3-4 meters.
Snjevci is an interesting island with stairs and a path that climbs to the top of it where view towards Elafiti Islands and towards Prožura opens up.