Fourier phase retrieval aims to reconstruct images from magnitude-only measurements; however, the absence of phase information, noises, and channel ambiguities make the recovery of natural color images challenging. Although diffusion-prior-based methods impose strong image constraints, their recovery performance degrades under higher measurement noise, and repeated generative updates incur substantial computational overhead. To overcome these limitations, this paper proposes a novel method, termed FlowFPR, that integrates a pretrained flow-matching prior with the relaxed averaged alternating reflections (RAAR) algorithm for oversampled Fourier phase retrieval. In the basic version of FlowPR (Flow-RAAR-basic), a color image is estimated by repeatedly performing four steps at each outer iteration, where the flow-prior prediction step exploits single-step straight-line prediction, considerably reducing the generation cost. Nevertheless, Flow-RAAR-basic still suffers from two critical issues: (i) the predefined fusion weights, \(λ_n\), might lead to either ineffective or excessive correction, and (ii) the fixed iteration number used by RAAR algorithm does not allow adaptation to measurement noise. To address these two issues, we develop two improved algorithms, Flow-RAAR-adaptive and Flow-RAAR-controlled, respectively. Moreover, these two improved algorithms naturally lead to a joint variant, Flow-RAAR-joint, which combines their advantages. Extensive experiments on a natural face image dataset demonstrate the superior performance of the four proposed methods compared with other relevant competing methods, as well as their acceleration over two recent diffusion-based methods. In the noise setting, Flow-RAAR-joint performs best among the four proposed methods.

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