In this article we establish new second order necessary and suffi-cient optimality conditions for a class of control-affine problems with a scalar control and a scalar state constraint. These optimality conditions extend to the constrained state framework the Goh transform, which is the classical tool for obtaining an extension of the Legendre condition. We propose a shooting algorithm to solve numerically this class of problems and we provide a suffi-cient condition for its local convergence. We provide examples to illustrate the theory.