There was never an official Church teaching about round earth geocentrism, but this is what was taught at the universities. Since these were medieval Catholic institutions, it is reasonable to refer to that as Church support.
There was no consensus among the Church Fathers on the shape of the earth. Some believed it to be flat and some round.
If by "wider population" you mean the uneducated people, it is difficult to determine what they believed. They did not leave records about what they thought of the shape of the earth or if they thought about it at all. While I agree there is no evidence to show that they accepted the round earth, neither is there evidence they believed it to be flat.
It is not reasonable at all to refer to this as church support. Errors push themselves in slowly. It would have crept in by the excuse that it was the domain of science. There is no evidence to show the the magesterium teaching round earthism ever. There is a difference between what was taught in some universities by some professors, and what Rome was teaching.
The majority of Fathers believed the Earth to be flat.
It is important as to what most ordinary people thought. All the evidence shows they did not accept the globe. Here are two examples which stand out
http://flatearthtrads.forumga.net/t145-hereford-cathedral-map-of-the-worldhttp://flatearthtrads.forumga.net/t141-hieronymus-bosch-15th-century-painter-flat-earth-paintingThe latter is from the 15th century.
If you are tempted to scoff, remember that on such an issue, these kind of things are regarded as important evidence to judge what the popular opinion was. Historians are like detectives and deal with whatever evidence they have, even if it is sparse.
On the science of the flat earth....