In this work, a nonlinear subdiffusion initial-boundary value problem with a variable exponent is considered, whose solution behaves a weak singularity at initial time. By adding a corrected term to the nonuniform L1 scheme, a novel scheme is investigated to approximate the time-fractional Caputo derivative with a variable exponent. This scheme allows us to use a smaller grading parameter $r$ to obtain a similar level of accuracy as that of the L1 method. Combining the proposed scheme with the finite element method in space and the Newton linearization for the nonlinear term, a fully discrete scheme is constructed. To obtain the unconditional optimal error estimate, the temporal-spatial splitting technique is adopted to derive the boundedness of the computed solution $U^n_h$ in $L^∞$-norm. With the help of this bound and the discrete fractional Gronwall inequality, the optimal error analysis without certain temporal restrictions dependent on the spatial mesh size is derived. Furthermore, by using a simple postprocessing technique of the computed solution, the convergence order in the spatial direction is improved. Finally, numerical experiments are presented to verify the theoretical findings.