WASHINGTON — When Energy Secretary Rick Perry led an American delegation to the inauguration of Ukraine’s new president in May, he took the opportunity to suggest the names of Americans the new Ukrainian government might want to advise and oversee the country’s state-owned gas company.

Mr. Perry’s focus during the trip on Ukraine’s energy industry was in keeping with a push he had begun months earlier under the previous Ukrainian president, and it was consistent with United States policy of promoting anti-corruption efforts in Ukraine and greater energy independence from Russia.

But his actions during the trip have entangled him in a controversy about a pressure campaign waged by President Trump and his allies directed at the new Ukrainian president, Volodymyr Zelensky, that is at the center of the impeachment inquiry into Mr. Trump. That effort sought to pressure Mr. Zelensky’s government to investigate Mr. Trump’s rivals, including former Vice President Joseph R. Biden Jr., a leading contender for the Democratic presidential nomination to challenge Mr. Trump.

Mr. Perry’s trip raised questions about whether he was seeking to provide certain Americans help in gaining a foothold in the Ukrainian energy business at a time when the new Ukrainian government was looking to the United States for signals of support in its simmering conflict with Russia.