SÃO PAULO—Brazil’s Supreme Court delivered a blow to President Jair Bolsonaro Friday by allowing the airing of a video of a cabinet meeting in which he appears to call on his then-justice minister to interfere in criminal investigations of family members.

The former minister, Sérgio Moro, who resigned last month after tangling with the president, had publicly said that the video would show that Mr. Bolsonaro pressured him to change the head of the Federal Police in Rio de Janeiro state to protect his sons.

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