WASHINGTON—Secretary of State Mike Pompeo faced an impatient and critical panel of senators at a hearing Wednesday, drawing criticism for the Trump administration’s handling of talks with Russia and North Korea and for other top foreign-policy initiatives.

Sen. Bob Corker (R., Tenn.), chairman of the Senate Foreign Relations Committee, opened the hearing by assailing President Trump for failing to disclose agreements reached with President Vladimir Putin, undermining the North Atlantic Treaty Organization and doling out aid...