The candidates started 'slinging mud' (at each other) early in the campaign.
His shoes were covered with 'mud'. = His shoes were caked in/with 'mud'.
The scandal ruined his reputation and now 'his name is mud'.
(..) Look out, fellows. Come on, get these papers out of here. Read all about it! Jeff Smith lies in Senate! Are we going to let a man like that throw mud at a man like Joe Paine? Are you for Joe Paine? Hurrah for Jeff Smith! Children hurt all over the city! Tell Jeff to stop! Yes, all right. (..)1939 Mr. Smith Goes to Washington