Kim Jong Un boasted Wednesday that North Korea enters the new year on a surge of strength because of the elimination of “factionalist filth”—a reference to the young leader’s once powerful uncle, whose execution last month raised questions about Kim’s grip on power. Kim’s comments in an annual New Year’s Day message, which included a call for improved ties with Seoul but also a warning of a possible “nuclear catastrophe,” will be scrutinized by outside analysts and governments for clues about the opaque country’s intentions and policy goals. Already widespread worry about the country has deepened since Kim publicly humiliated...