Observations of reionization-era analogs at $zsim3$ are a powerful tool for constraining reionization. Rest-ultraviolet observations are particularly useful, in which both direct and indirect tracers of ionizing-photon production and escape can be observed. We analyse a sample of 124 $zsim3$ galaxies from the Keck Lyman Continuum Spectroscopic Survey, with sensitive spectroscopic measurements of the Lyman continuum region. We present a method of removing foreground contamination from our sample using high-resolution, multi-band Hubble Space Telescope imaging. We re-measure the global properties of the cleaned sample of 13 individually-detected Lyman continuum sources and 107 individually-undetected sources, including a sample-averaged absolute escape fraction of $f_{rm esc,abs}=0.06pm0.01$ and a sample-averaged ratio of ionizing to non-ionizing ultraviolet flux density of $<f_{900}/f_{1500}>_{rm out}=0.040pm0.006$, corrected for attenuation from the intergalactic and circumgalactic media. Based on composite spectra, we also recover a strong positive correlation between $<f_{900}/f_{1500}>_{rm out}$ and Ly$alpha$ equivalent width (W$_lambda$(Ly$alpha$)) and a negative correlation between $<f_{900}/f_{1500}>_{rm out}$ and UV luminosity. As in previous work, we interpret the relationship between $<f_{900}/f_{1500}>_{rm out}$ and W$_lambda$(Ly$alpha$) in terms of the modulation of the escape of ionizing radiation from star-forming galaxies based on the covering fraction of neutral gas. We also use a W$_lambda$(Ly$alpha$)-weighted $<f_{900}/f_{1500}>_{rm out}$ to estimate an ionizing emissivity from star-forming galaxies at $zsim3$ as $epsilon_{rm LyC}simeq5.5times10^{24}$erg s$^{-1}$ Hz$^{-1}$ Mpc$^{-3}$. This estimate, evaluated using the uncontaminated sample of this work, reaffirms that galaxies provide the majority of the ionizing background at $zsim3$ and beyond.
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