Unstructured finite volume methods are typically categorized based on control volumes into the node- and cell-centered types. Because of certain inherent geometric properties, the second-order Linear-Galerkin scheme, favored for its simplicity and ability to preserve second-order solution accuracy, is predominantly applied to node-centered control volumes. However, when directly applied to cell-centered control volumes, the designed solution accuracy can be lost, particularly on irregular grids. In this paper, the least-square based Linear-Galerkin discretization is extended to arbitrary cell-centered elements to ensure the second-order accuracy can always be achieved. Altogether four formulations of corrected schemes, with one being fully equivalent to a conventional second-order finite volume scheme, are proposed and examined by problems governed by the linear convective, Euler and Navier-Stokes equations. The results demonstrate that the second-order accuracy lost by the original Linear-Galerkin discretization can be recovered by corrected schemes on perturbed grids. In addition, shock waves and discontinuities can also be well captured by corrected schemes with the help of gradient limiter function.