The article is devoted to the study of linear inverse problems of finding, alongside the solution, also the unknown right-hand side for second-order differential equations. The method of the study is based on reducing the initial inverse problems to direct, already nonlocal, boundary value problems for loaded (integro-differential) equations, proving the solvability of the new problems and then constructing solutions to the problems under study from the solutions to the new problems. The peculiarities of the inverse problems under study are new overdetermination conditions compared to the previous works. For the problems under study, we prove existence theorems for regular solutions (i.e. for solutions having all weak derivatives in the sense of Sobolev occurring in the equation). Some generalizations and strengthenings of the obtained results are given.