A QPR youth, Goddard was West Ham's record signing when he moved to Upton Park for £800,000 in 1980. He was West Ham's top scorer in the 1982–83 season, and scored 71 goals in 213 League and Cup matches for the club. One of these goals was in the replay of the 1981 Football League Cup Final defeat to Liverpool. At the start of the 1985-86 season he partnered Tony Cottee in the West Ham front line, but injury ruled him out after one game. This led to Cottee being partnered by Frank MacAvennie, and the MacAvennie-Cottee partnership yielded 54 goals in one season, taking West Ham to 3rd place in the league.
He then moved to Newcastle United in October 1986. He then moved on to Derby County in 1988 before signing for Millwall for a club record transfer fee in December 1989.
In 1991, he moved to Ipswich Town on a free transfer and enjoyed a few more years in the Premier League before retiring in 1994. During the 1994/95 season, he was temporarily caretaker-manager with old teammate John Wark between the departure of John Lyall and the arrival of George Burley and went on to become the club's youth team coach.
Later he worked as assistant manager at West Ham United for Glenn Roeder, a former teammate when he played at QPR and Newcastle United. He left that post on 20 January 2004, shortly after the arrival of new manager Alan Pardew.
As of 2008, Paul lives with his family in East Bergholt, Suffolk and since 2005 has worked for the Stellar Group for their football agency.
