This　paper　focuses　on　efficient　projection　onto　the　intersection　of　a　half-space　and　a　box-like　set　and　its　generalized　Jacobian.　Based　on　the　Lagrangian　duality　theory,　we　deal　with　the　projection　problem　via　a　semismooth　Newton　algorithm　with　line　search　safeguard,　which　admits　global　and　locally　quadratic　convergence,　to　solve　a　univariate　semismooth　equation.　Numerical　experiments　show　that　our　proposed　algorithm　outperforms　favourably　the　existing　state-of-the-art　standard　solvers　and　is　able　to　reliably　solve　very　large-scale　projection　problems.　Besides,　we　derive　an　explicit　expression　of　a　generalized　Jacobian　of　the　studied　projection,　which　is　an　essential　component　of　second-order　nonsmooth　methods.