Recovery of block sparse signals with partially-known block support information is of particular importance in compressed sensing. A uniform sufficient condition guaranteeing stable recovery of non-strictly block $k$-sparse signals is established via the weighted $ℓ_{2,p}−\alpha ℓ_{2,q}$ nonconvex minimization method, and the reconstruction error is precisely bounded in terms of the residual of block-sparsity and the measurement error. Furthermore, a series of contrastive numerical experiments reveal that exploiting the approximate block-sparsity characteristic and the nonuniform prior block support estimate substantially promotes the performance of reconstruction for block-structural signals.