Cu-C materials and interfaces are central to diamond thermal management, metallization, graphene growth and carbon-coating technologies, but atomistic simulations across these applications require a potential that describes metallic Cu, diverse carbon bonding, heterogeneous Cu-C interactions and close atomic encounters. Here we develop NEP-CuC, a transferable neuroevolution potential trained on a dataset of 2,727 uniformly relabelled configurations selected from a heterogeneous Cu-C candidate pool. Disagreement within an ensemble of five independently trained models provides an internal indicator of model sensitivity. A Ziegler-Biersack-Littmark (ZBL) correction describes short-range nuclear repulsion, while a runtime D3(BJ) correction supplements dispersion-sensitive calculations. Tests on source-derived held-out and independent external datasets, together with structural, elastic, defect, interfacial, vibrational and finite-temperature transport benchmarks, establish broad but property-dependent physical fidelity across elemental Cu, diverse carbon environments and heterogeneous Cu-C configurations. Large-scale simulations of amorphous-carbon formation and orientation-dependent shock compression of Cu further test transferability under strongly non-equilibrium conditions. For Cu/diamond interfaces, the same potential describes deposition, annealing and tensile response: preparation history leaves persistent differences in interphase structure, while the mean peak nominal tensile stress at 500 K decreases from 12.35±0.71 to 6.65±0.98 GPa as the incident Cu energy during the initial deposition stage increases from 1 to 50 eV. NEP-CuC provides an atomistic framework for linking processing, interfacial structure and physical response across Cu-C systems, with direct relevance to Cu/diamond thermal-management interfaces, diamond metallization and non-equilibrium Cu-C processing.