I'm pretty sure supply production and distribution works the same in all of the AGEod games.
Small cities generally produce very little to no supply, although some might have some kind of factory, arsenal, or such, but generally those are in larger cities.
Regardless, your forces practically never live off the supplies created in a single city, but through the supplies produced from all over the nation and transported through depots by rail and river to the front.
For supply to move through a region, you must have at least 25% MC in that region. For the city in a region to produce resources, you must control the city, generally by being the last one to have had a non-artillery combat unit inside the city, or an unopposed non-artillery combat unit in the region--although this doesn't work with early war cavalry, who only capture a city if you have >50% loyalty in the region. So to capture cities in the South, where you have very little loyalty, as the Union you need late war cavalry or at least a militia unit.
For a better understanding of how supplies are moved, please review
Supply Primer in the old AACW forum.
Management Summery
You need a network of depots to move supply. The depots should be connected to each other by rail lines and/or rivers for depots in cities with harbors, and not too far apart--generally 3 regions at the most if connected by rail or river, 2 if only by roads.