Quantum error correction will be an essential ingredient in realizing fault-tolerant quantum computing. However, most correction schemes rely on the assumption that errors are sufficiently uncorrelated in space and time. In superconducting qubits, this assumption is drastically violated in the presence of ionizing radiation, which creates bursts of high-energy phonons in the substrate. These phonons can break Cooper pairs in the superconductor and, thus, create quasiparticles over large areas, consequently reducing qubit coherence across the quantum device in a correlated fashion. A potential mitigation technique is to place large volumes of normal or superconducting metal on the device, capable of reducing the phonon energy to below the superconducting gap of the qubits. To investigate the effectiveness of this method, we fabricate a quantum device with four nominally identical nanowire-based transmon qubits. On the device, half of the niobium-titanium-nitride ground plane is replaced with aluminum (Al), which has a significantly lower superconducting gap. We deterministically inject high-energy phonons into the substrate by voltage biasing a galvanically isolated Josephson junction. In the presence of the small-gap material, we find a factor of 2-5 less degradation in the injection-dependent qubit lifetimes and observe that the undesired excited qubit state population is mitigated by a similar factor. We furthermore turn the Al normal with a magnetic field, finding no change in the phonon protection. This suggests that the efficacy of the protection in our device is not limited by the size of the superconducting gap in the Al ground plane. Our results provide a promising foundation for protecting superconducting-qubit processors against correlated errors from ionizing radiation.