In this paper, the controllability of second-order problems in Banach spaces is investigated when the nonlinear term also depends on the first derivative. The main aim of the paper is to introduce the definition of controllability for second-order problems in Banach spaces that considers both the solution and its derivative at the final point using a unique control and to obtain sufficient conditions for such controllability. Our main results are derived by combining the Schauder fixed point theorem with the approximation solvability method and weak topology. This approach allows us to obtain results under easily verifiable and non-restrictive conditions imposed on the cosine family generated by the linear operator and on the right-hand side since any requirements for compactness are avoided. The paper concludes by applying the obtained results to a system governed by the one-dimensional Klein-Gordon equation.