Among all plasmonic metals, copper (Cu) has the greatest potential for realizing optoelectronic and photochemical hot-carrier devices, thanks to its CMOS compatibility and outstanding catalytic properties. Yet, relative to gold (Au) or silver (Ag), Cu has rarely been studied and the fundamental properties of its photo-excited hot carriers are not well understood. Here, we demonstrate that Cu nanoantennas on p-type gallium nitride (p-GaN) enable hot-hole-driven photo-detection across the visible spectrum. Importantly, we combine experimental measurements of the internal quantum efficiency (IQE) with ab initio theoretical modeling to clarify the competing roles of hot-carrier energy and mean-free path on the performance of hot-hole devices above and below the interband threshold of the metal. We also examine Cu-based plasmonic photodetectors on corresponding n-type GaN substrates that operate via the collection of hot electrons. By comparing hot hole and hot electron photodetectors that employ the same metal/semiconductor interface (Cu/GaN), we further elucidate the relative advantages and limitations of these complementary plasmonic systems. In particular, we find that harnessing hot holes with p-type semiconductors is a promising strategy for plasmon-driven photodetection across the visible and ultraviolet regimes. Given the technological relevance of Cu and the fundamental insights provided by our combined experimental and theoretical approach, we anticipate that our studies will have a broad impact on the design of hot-carrier optoelectronic devices and plasmon-driven photocatalytic systems.