Measuring part of a quantum system in the midst of its evolution and acting on the result in real time is essential for numerous quantum information protocols. Neutral-atom arrays are a leading platform for quantum information processing, but their mid-circuit measurement-and-feedforward cycle times have remained slow, typically exceeding 1 ms. Here we demonstrate fast mid-circuit measurement and real-time feedforward in an array of atomic qubits coupled to a high-finesse optical cavity. Local light shifts tune individual data qubits out of resonance with the cavity, shielding their coherence, while a near-resonant probe drives a selected qubit whose emission is collected with Purcell enhancement. Mid-circuit measurements of four qubits with subpercent infidelity reduce the coherence of a fifth unmeasured data qubit by less than 2%. We implement real-time feedforward to correct measurement-induced phase shifts and to realize an adaptive circuit for optimal quantum-state discrimination and conditional state preparation. Our approach reduces the measurement-and-feedforward cycle time to below 100 µs and establishes optical cavities as a route to fast control of neutral-atom quantum systems.