This paper examines the controllability of the second-order evolution problems in Banach spaces in the presence of a nonlinear term depending on both the solution and its first derivative, and of a time-dependent linear term. The first goal of the paper is to introduce a definition of controllability for such problems that considers not only the solution but also its derivative at the final point using the same control function. Subsequently, the paper addresses the second goal: Finding conditions that guarantee the solvability of the controllability problem. The main statements of the paper are proved by the combination of Schauder fixed point theorem, approximate solvability, and weak topology. Using the method of approximate solvability allows us to avoid any compactness restrictions, and the results are therefore proved under non restrictive conditions imposed on the fundamental operator as well as on the right-hand side.