Ex-Trump campaign adviser George Papadopoulos wasn’t helpful to the special counsel’s investigation, hurt investigators’ efforts to detain a Russian intermediary and should face at least one month in prison, special counsel Robert Mueller said in a filing late Friday.

Mr. Papadopoulos’s meetings with a Russian conduit, a professor in London who told him about Russian “dirt” on Hillary Clinton, led to the launch of the U.S. counterintelligence investigation into Russian interference in the 2016 election.

Mr....