The Physical Random Access Channel (PRACH) is essential for initial access and synchronization in both 5G and future 6G networks; however, its detection is highly sensitive to impairments such as high user density, large carrier frequency offset (CFO), and fast fading. Although prior studies have examined PRACH detection, they are often restricted to specific scenarios or lack a comprehensive analytical characterization of performance. We introduce a unified analytical framework that characterizes the statistical distribution of the received power delay profile (PDP) under flat Rayleigh fading and supports both coherent combining (CC) and power combining (PC) repetition strategies. For each strategy, we derive optimal threshold expressions and closed-form detection probabilities. Furthermore, we analyze two key cases depending on the coherence time: identical and independent channel realizations per repetition. Secondly, we exploit the correlation induced by CFO across cyclic shifts to design a novel low-complexity detector that exploits PDP dependencies. Numerical results indicate that PC outperforms CC when repetitions experience independent channels, while CC can be preferable under identical realizations in limited settings. On the other hand, the proposed CFO-aware detector delivers improved robustness under severe CFO conditions.