In this note, we extend our recent work, which studied high-order nonuniform method for a time-fractional nonlinear PDE, to construct a graded linearized scheme for a time-fractional Boussinesq type equation which involved a Caputo derivative of order β∈(1,2) in time, when the solution is not smooth. We show that the numerical solution converges to the exact solution with temporal third-order accuracy for fitted grading parameter bounded in (1,1.5). This result is different from previous ones where gradings will tend to infinity.