The size of Copernicus’ and Ptolemy’s corrections I understand that both Copernicus and Ptolemy needed epicycles, but one might assume that a heliocentric system would need much smaller corrections to account for the non-circular motion of the planets.  This is however not mentioned as an important reason to favor Copernicus’ system. Is this not the case?
I gather the worse problem for Copernicus was Mercury.  Not only is Mercury's orbit quite elliptical, it has relatively high precession due to the gravitational influence of other planets.

I found a quote attributes to Copernicus in this document (sorry, only direct link available) where he says :


  It is interesting to note that Copernicus had difficulty explaining Mercury’s motion and once commented that ”this planet has .  ..  influenced many perplexities and labours on us in our investigation of its wanderings".


This is apparently originally from this source :


Baum, Richard and Sheehan, William.In Search of Planet Vulcan:  The Ghost in Newton’s Clockwork Universe. Plenum Trade, New York. 1997.


The gravitational perturbation of the planets, particularly by Jupiter, and the concept of barycenter for even ideal elliptical orbits required the work of Newton (following Kepler's concept of elliptical orbits) to explain.  Up until Newton there was no solid theory for the motion at all and people were "just" (poor choice of words) trying to find the simplest model.  Copernicus provided a simple model, but also a less accurate one.