Cell-free network and intelligent reflecting surface (IRS) are considered as two promising technologies for improving future network capacity and coverage. They offer advantages such as low cost, low energy consumption, and compliance with green communication requirements. However, the static deployment of IRS restricts the network's ability to adapt to emergency coverage requirements and dynamic environments. To address this issue, we propose a flexible IRS-aided cell-free network, where network capacity and signal coverage are substantially improved by utilizing reflected signals from an aerial IRS. Our objective is to maximize the weighted transmission rate for users by jointly optimizing the active beamforming of the base stations (BSs), passive beamforming of the IRS, and the location of the unmanned aerial vehicle (UAV). Due to the non-convex and intractable nature of this problem, we decompose it into three subproblems. For the optimization problems of BSs' active beamforming and IRS's passive beamforming, we transform the log-sum problem into a quadratically constrained quadratic programming (QCQP) problem by employing the lagrangian dual principle and multi-ratio fractional programming. Furthermore, for the more challenging location optimization, we further transform it into a convex problem using the successive convex approximation (SCA) technique to obtain a high-quality suboptimal solution. Simulation results demonstrate that the proposed scheme can significantly improve the weighted transmission rate and effectively enhance the network coverage as compared to other benchmarks.