WASHINGTON — Rep. Katherine Clark of Massachusetts, the No. 2 House Democrat, is set to become the majority leader in January, after her party won control of the chamber.

Clark, 63, a quiet, progressive power player in Congress, would make history as the first woman to hold that role, though Rep. Nancy Pelosi broke a higher glass ceiling in 2007 when she became the first woman to be speaker.

“I feel a weight of responsibility,” Clark said in an interview this week. “It’s the women’s voices from traveling across my district and the country that really haunt me.”

Described by friends as similar in ideology and temperament to Sen. Elizabeth Warren, also of Massachusetts, Clark is set to be the most liberal member on House Democrats’ leadership team next year.…