We consider the local error estimate of the L1 scheme on graded mesh for a linearized time fractional KdV equation with weakly singular solutions, where Legendre Petrov–Galerkin spectral method is used for the spatial discretization. Stability and convergence of the fully discrete scheme are rigorously established, and the pointwise-in-time error estimates are given to show that one can attain the optimal convergence order 2− α in positive time by mildly choosing the grading parameter r no more than 2 for all 0< α< 1. Numerical results are presented to show that the error estimate is sharp.