TOKYO—Prime Minister Yoshihide Suga said he won’t seek re-election as ruling-party leader, effectively ending his term after just a year and bringing back political instability in one of Washington’s most important allies.

Japan’s recent wave of Covid-19 cases caused by the Delta variant did irreparable damage to Mr. Suga’s leadership, and his support rating in recent polls fell below 30%. Public dissatisfaction rose even after a relatively problem-free Tokyo Olympics, which included a record medal haul for the home team.

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