SEOUL, South Korea — The North Korean leader, Kim Jong-un, on Wednesday called for improving relations with South Korea and boasted of his regime’s tightened grip on power in his first public speech since the purge and execution of his uncle, Jang Song-thaek, last month.

“North and South Korea should create a mood to improve relations,” Mr. Kim said in a nationally televised New Year’s Day speech. “It’s time to end useless slandering, and the North and the South should no longer do things that harm reconciliation and harmony.”

Mr. Kim began delivering a New Year’s Day speech after coming to power two years ago, reviving the practice of his grandfather Kim Il-sung. During the rule of his reclusive father, Kim Jong-il, the country’s main state-run newspapers issued a joint editorial to mark the day.

Analysts waited for this year’s speech with unusual interest because it would be the first since the purge of Mr. Jang, long considered Mr. Kim’s mentor and the regime’s No. 2 figure. Mr. Jang was executed on Dec. 12 on charges of building a faction within the ruling Workers’ Party in a plot to overthrow Mr. Kim’s government.