Kill Leonine Misbegotten
To complete this square you need to defeat Leonine Misbegotten, the boss of Castle Morne. Wait until the "ENEMY DEFEATED" or equivalent screen appears to mark the square.
Castle Morne has a long travel time, which makes the square more unattractive. On the other hand, Leonine is an early game boss, easier to defeat than other options available from the start.
Leonine Misbegotten counts as a Limgrave/Weeping Peninsula boss.
DANGER
No other Misbegotten boss counts for this square; you need to kill the Castle Morne boss.