Ukraine lost Kherson within the first few weeks of the war. Russia took the city relatively easier than other cities in eastern Ukraine. There's some evidence that the local officials in Kherson simply switched sides early on which is part of why the city was captured by Russia so quickly.
As for the Russian withdrawal, it happened because Russia's commanders wanted to leave Kherson city. After Ukraine was given HIMAR launchers, it became too difficult for Russia to supply their forces on the west side of the river in Kherson. So it's more accurate to say that Russia surrendered the city due to the logistics problems. They weren't driven out by Ukrainian ground forces (who failed in dozens of attempts take Kherson between the summer and fall).
Soledar and Bakhmut reportedly are major logistics hubs for Ukrainian forces. Losing those cities will make it more difficult for Ukraine's supply lines on the eastern front. There are also reportedly miles of underground tunnels in that area which the Russians can use to support their logistics in Ukraine and hide equipment from spy western satellites.
The battles for those two cities have been costly for both sides. I doubt that Ukraine would sacrifice so many troops to defend them if they were insignificant targets.
Kherson would not have been hard to keep supplied, if it was not being effectively attacked by Ukraine. So, yes, the Russians were driven out of Kherson.
The Ukrainians took an even larger slice of territory back near Kharhiv. Was this territory on the wrong side of a river?
How are the Russians going to hold the Crimea when it's not on the wrong side of a river but on the wrong side of a strait, with just one long bridge across it, and a narrow and vulnerable land bridge connection? That will be a neat trick.
The Ukrainians have already taken back half of the territories the Russians took in the first few months of the war in 2022. Look for more of the same in 2023, when the ground dries out. And the Ukrainian soldiers who are now receiving NATO training on combined arms return to the front. While reluctant Russian draftees receive almost no training and have their winter clothes stolen from them.
How many of the 24 Ukrainian Oblast administration centers did Russia take in 2022? Only one, Kherson. How many of these does Russia still control? Zero.
Any city of 70,000 or 10,000 could be called "a major communication hub". Name me one city that isn't?
Soledar is strategic because of it's tunnels? It could be used to store ammunition? Yes, it would be a great place. A huge ammunition dump right on the front line, where a minor advance by Ukraine allows the Ukrainian army to capture it and use the ammunition themselves. And how about the job of a Russian truck driver. Driving a Russian truck filled with artillery ammunition right up to the Ukrainian front line. Yes, this makes total sense. I can see how taking Soledar would be an immediate strategic victory for Russia that they can start taking advantage of right away. Soledar won't be useful until Soledar is taken and the Russians are able to drive the Ukrainians back many more miles from there, something we don't see anything like that remotely happening.
Bakhmut and Soledar are strategically important because they happen to lie on the Ukrainian side of the front where it ended up being established when Ukraine managed to stop the Russian offensives this year. Nothing more. The Russians are just looking for some sort of victory to present to the Russian people to offset the major loss of territory in the Kharkiv and Kherson regions.
