In this paper, we study the inverse problem of finding a time-dependent multiplier of the right-hand side of a time-fractional one-dimensional diffusion equation with variables coefficients in the case where the usual Cauchy, homogeneous Dirichlet boundary, and an integral overdetermination conditions are given. The overdetermination condition has the form of an integral with a weight over a spatial segment from the solution of the direct problem, in which the weight function is a spatially dependent known factor of the right-hand side of the equation. This made it possible to construct a solution to the inverse problem in explicit form and prove its correctness in the class of regular solutions