Wayne Rooney won just four out of 23 league games in charge of Plymouth Argyle, which has seen them sit four points off safety in the EFL Championship.
Chelsea and West Ham progressed to the FA Cup fifth round on Saturday after identical 2-1 wins in extra time.
Liverpool, third in the Premier League, will host second-tier Wolverhampton Wanderers in the FA Cup round four on January 28.
Tottenham Hotspur joined Chelsea in round four by beating Aston Villa 2-0, but Liverpool face a replay after being held 0-0 by Plymouth