Predictor antenna (PA) system is referred to as a system with two sets of antennas on the roof of a vehicle, where the PAs positioned in the front of the vehicle are used to predict the channel state observed by the receive antennas (RAs) that are aligned behind the PAs. In this work, we study the effect of spatial mismatch on the accuracy of channel state information estimation, and analyze the delay-constrained average rate of hybrid automatic repeat request (HARQ)-based PA systems. We consider variable-length incremental redundancy (INR) HARQ protocol, and derive a closed-form expression for the maximum average rate subject to a maximum delay constraint. Moreover, we study the effect of different parameters such as the vehicle speed, the antenna distance and the rate allocation on the system performance. The results indicate that, compared to the open-loop case, the delay-limited average rate is considerably improved with our proposed PA-HARQ scheme.