TITLE: ``Hubble Space Telescope Observations of Escaping Lyman Continuum Radiation from Galaxies and Weak AGN at Redshifts 2.3 ~= 2.37, 2.67, 3.50, and ~5.1 is generally detected at the >3--4-sigma level, in typical image stacks of 11--37 objects in the WFC3/UVIS F225W, F275W, F336W, and ACS/WFC F435W filters, respectively. Their measured signal corresponds to total LyC fluxes of AB~29--31 mag. The LyC flux of weak AGN is detected at ~1 mag brighter levels at z~=2.3--4.8, but averaged over 4--10x fewer objects per stack. The stacked galaxy LyC light-profiles are much flatter than their UV-continuum profiles at radii r=<0.7". Their weighted-average LyC light-profile is significantly non-centrally concentrated over this redshift range, which may be explained by a radial dependence in the ISM porosity at 2.3 < z < 5. With SED-fitting of the UV-continuum longwards of Lya, the observed LyC flux corresponds to an average LyC escape fraction of f_esc=7--50% at z=2.3--3.5, while increasing to possibly ~100% at z>4.35. The available data for galaxies are bounded by a trend with redshift of f_esc = (0.02 +/- 0.01) x (1+z)^{1.0+/-0.5}. However, a better description seems to be a more sudden increase in f_esc with redshift that occurred around z = 2.9, which we suggest is due to dust accumulating in galaxies as they grow with cosmic time. Our best fits of f_esc(z) suggest that faint galaxies collectively may have measurably contributed to maintaining cosmic reionization at redshifts z>3--5, while (weak) AGN likely produced a significant fraction of the ionizing LyC flux at z ~< 2.5. Complete reionization at z<6--7 may be further enabled by a steep faint-end of the galaxy luminosity function at higher redshifts, integration to very faint luminosities (M_AB = -14 mag), and possibly also that f_esc increases at fainter luminosities and lower metallicities at higher redshifts than can be sampled with the current data.