The Patriots released Nugent after the team's 27-13 win over the Browns on Sunday. He missed a 34-yard field goal and had a 29-yard attempt blocked. It was the first blocked field goal against New England since 2006.
New England originally signed Nugent, 37, on Oct. 3 after longtime kicker Stephen Gostkowski was placed on season-ending injured reserve due to a hip injury. In four games with the Patriots, he was 5-for-8 on field goals and 15-of-16 on extra points.
Folk last kicked for the Alliance of American Football's Arizona Hotshots. The 34-year-old kicker last played in the league with the Tampa Bay Buccaneers in 2017.
The Patriots also announced that tight end Eric Tomlinson, who started two games, was released. He was signed on Oct. 15 and recorded one reception for one yard.
New England will play the Baltimore Ravens on Sunday Night Football.