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We theoretically study the effect of external deformation on activated structural relaxation and elementary aspects of the nonlinear mechanical response of glassy hard sphere fluids in the context of the nonequilibrium version of the Elastically Collective Nonlinear Langevin Equation (ECNLE) theory. ECNLE theory describes activated relaxation as a coupled local-nonlocal event involving local caging and longer range collective elasticity, with the latter becoming more important with increasing packing fraction. The central new question is how this physical picture, and the relative importance of caging versus elasticity physics, depends on external stress, strain and shear rate. Theoretical predictions are presented for deformation induced enhancement of mobility, onset of relaxation speed up at remarkably low values of stress, strain or dimensionless shear rate, thinning of the structural relaxation time and viscosity with apparent power law exponents, a non-vanishing activation barrier in the shear thinning regime, a Herschel-Bulkley form of rate dependence of the steady state shear stress, exponential growth of dynamic yield stresses with packing fraction, and reduced dynamic fragility and heterogeneity under deformation. The results are contrasted with experiments and simulations, and qualitative or better agreement is found. An overarching conclusion is that deformation strongly reduces the importance of longer range collective elastic effects for most, but not all, physical questions, with stress-dependent dynamic heterogeneity phenomena being qualitatively sensitive to collective elasticity. Overall, nonlinear rheology is a more local cage scale problem than quiescent relaxation, albeit with deformation-modified activated processes still important.