Considering the one-gluon exchange interaction and phenomenological quark confinement potential, an improved light-cone effective Hamiltonian for mesons and the corresponding radial mass eigen equations in angular momentum representation is obtained. Solving the J = 0 eigen equations numerically and using a set of adjustable parameters, the obtained solutions for ground states and radial excited states can simultaneously describe both pseudoscalar and scalar flavour-off-diagonal mesons. Some radial excited states are also predicted and wait for experimental test. More results for the vector and axial vector mesons are expected.