WASHINGTON—House Democratic leaders are signaling a renewed emphasis on health care and other issues over opposition to President Trump even as they press for the release of special counsel Robert Mueller’s report about obstruction and Russian election interference.

“We’ve never taken our focus off the for-the-people agenda,” House Speaker Nancy Pelosi (D., Calif.) told reporters on Tuesday after meeting with the Democratic caucus. She said the party would push to lower the cost of health care, build infrastructure and clean...