In this article, an $α$-th$ (0<α<1)$ order time-fractional reaction-diffusion equation with variably diffusion coefficient and initial weak singularity is considered. Combined with the fast $L1$ time-stepping method on graded temporal meshes, we develop and analyze a fourth-order compact block-centered finite difference (BCFD) method. By utilizing the discrete complementary convolution kernels and the $α$-robust fractional Grönwall inequality, we rigorously prove the $α$-robust unconditional stability of the developed fourth-order compact BCFD method whether for positive or negative reaction terms. Optimal sharp error estimates for both the primal variable and its flux are simultaneously derived and carefully analyzed. Finally, numerical examples are given to validate the efficiency and accuracy of the developed method.