Clean materials are required to construct and operate many low-background physics experiments. High-purity copper has found broad use because of its physical properties and availability. In this paper, we describe methods to assay and mitigate $^{210}$Pb contamination on copper surfaces, such as from exposure to environmental radon or coming from bulk impurities. We evaluated the efficacy of wet etching on commercial samples and observed that $^{210}$Po contamination from the copper bulk does not readily pass into solution. During the etch, the polonium appears to trap at the copper-etchant boundary, such that it is effectively concentrated at the copper surface. We observed a different behavior for $^{210}$Pb; high-sensitivity measurements of the alpha emissivity versus time indicate the lowest level of $^{210}$Pb contamination ever reported for a commercial copper surface: $0pm12$ nBq/cm$^2$ (1$sigma$). Additionally, we have demonstrated the effectiveness of mitigating trace $^{210}$Pb and $^{210}$Po surface backgrounds using custom, high-purity electroplating techniques. These approaches were evaluated utilizing assays performed with an XIA UltraLo-1800 alpha spectrometer.
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