In this work, we present basic results and applications of Stepanov pseudo-almost periodic functions with measure. Using only the continuity assumption, we prove a new composition result of mu-pseudo-almost periodic functions in Stepanov sense. Moreover, we present different applications to semilinear differential equations and inclusions with weak regular forcing terms in Banach spaces. We prove the existence and uniqueness of mu-pseudo-almost periodic solutions (in the strong sense) to a class of semilinear fractional inclusions and semilinear evolution equations respectively, provided that the nonlinear forcing terms are only Stepanov mu-pseudo-almost periodic in the first variable and not a uniformly strict contraction with respect to the second argument. Our results are obtained using the Meir-Keeler principle and the Banach fixed point principle respectively. Some examples of fractional and nonautonomous partial differential equations illustrating our theoretical results are also presented.