Andrews underwent surgery for a broken thumb on his snapping hand and could be sidelined for additional games, according to Fox Sports' Jay Glazer.
The five-year veteran practiced snapping with his non-dominant hand in a bid to be active for Week 3, but to no avail.
Andrews missed the entire 2019 season after being diagnosed with a pulmonary embolism. His Week 1 start against the Miami Dolphins this season was his first game since the Patriots' Super Bowl LIII win.
The former Georgia product has started 59 games since signing with New England as an undrafted free agent in 2015.