In this paper, we consider a variation of the minimum-cost network flow problem (NFP) with additional non-convex cycle constraints on nodal variables; this problem has relevance in the context of optimizing power flows in electric power networks. We propose one approach to tackle the NFP that relies on solving a convex approximation of the problem, obtained by augmenting the cost function with an entropy-like term to relax the non-convex constraints. We show that the approximation error, for which we give an upper bound, can be made small enough for practical use. An alternative approach is to solve the classical NFP, i.e., without the cycle constraints, and solve a separate optimization problem afterwards in order to recover the actual flows satisfying the cycle constraints; the solution of this separate problem maximizes the individual en- tropies of the cycles. Finally, we validate the practical usefulness of the theoretical results through a numerical example, where the IEEE 39-bus system is used as a test bed.