We present observations of escaping Lyman Continuum (LyC) radiation from 34 massive star-forming galaxies and 12 weak AGN with reliably measured spectroscopic redshifts at $z$$simeq$2.3-4.1. We analyzed Hubble Space Telescope Wide Field Camera 3 (WFC3) mosaics of the Early Release Science field in three UVIS filters to sample the rest-frame LyC over this redshift range. With our best current assessment of the WFC3 systematics, we provide 1$sigma$ upper limits for the average LyC emission of galaxies at $z$=2.35, 2.75, and 3.60 to $sim$28.5, 28.1, and 30.7 mag in image stacks of 11-15 galaxies in the WFC3/UVIS F225W, F275W, and F336W, respectively. The LyC flux of weak AGN at $z$=2.62 and 3.32 are detected at 28.3 and 27.4 mag with SNRs of $sim$2.7 and 2.5 in F275W and F336W for stacks of 7 and 3 AGN, respectively, while AGN at $z$=2.37 are constrained to $gtrsim$27.9 mag at 1$sigma$ in a stack of 2 AGN. The stacked AGN LyC light profiles are flatter than their corresponding non-ionizing UV continuum profiles out to radii of r$lesssim$0.9, which may indicate a radial dependence of porosity in the ISM. With synthetic stellar SEDs fit to UV continuum measurements longwards of Ly$alpha$ and IGM transmission models, we constrain the absolute LyC escape fractions to $f_{rm esc}^{rm abs}$$simeq$$22^{+44}_{-22}$% at $z$=2.35 and $lesssim$55% at $z$=2.75 and 3.60, respectively. All available data for galaxies, including published work, suggests a more sudden increase of $f_{rm esc}$ with redshift at $z$$simeq$2. Dust accumulating in (massive) galaxies over cosmic time correlates with increased HI column density, which may lead to reducing $f_{rm esc}$ more suddenly at $z$$lesssim$2. This may suggest that star-forming galaxies collectively contributed to maintaining cosmic reionization at redshifts $z$$gtrsim$2-4, while AGN likely dominated reionization at $z$$lesssim$2.
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