With Ukraine’s reported crossing of the Oskil River including the taking of the eastern half of Kupiansk, the Russian defense of Lyman is now fatally flawed. Had the Russians been able to hold the Oskil, the river as defensive feature might have made up for the now incredibly poor logistics of the area for Russia. War-mapper has an excellent map of the larger area:
As you can see from their map, the rail lines north of Lyman and Kreminna both go to Kupiansk and therefore no supplies will be coming through from the north. From the south, the Lyman line goes through Ukrainian held Siversk and the Kreminna line has to cross the river between Severodonetsk and Lysychansk. If that railroad bridge is still up, the Ukrainians have the ability to bring it down any time they want. This puts Lyman at a roughly 60 km drive to a usable railroad. Historically, Russia has done poorly with truck supply at this distance.
Furthermore, there are multiple rumored crossings of the Oskil. Combined with recent Russian losses in the area and few places to draw reinforcement from, we can expect the Russian lines to be just as thin as they were just prior to the recent breakout in the area. When you factor in most of this line will have just as poor or worse supply than Lyman, they should be easy to break with a concerted Ukrainian push.
Ukraine can choose pretty much any point north of Lyman defended by a thin Russian force. A breakout there, plus pushing from Kupiansk down both the railroad on the east side of the Oskil plus the railroad towards Svatove will threaten to encircle the Russian forces in Lyman. Ukraine can use these rail lines relatively quickly to keep their forces in far better supply than the Russians. Svatove will not offer much resistance as it has the same supply problems that Lyman does.
This will hopefully be followed up with a drive on Starobilsk and then a right turn into Luhansk. Cutting the Starobilsk line plus getting close to Luhansk will put the eastern railroad into Luhansk in HIMARS range creating more problems along the whole front. At that point Russia will be supplying everything in the Donbas with just a single rail line from Rostov-on-Don. Everything West of Donetsk is not even connected to that rail line. I’m hoping getting to that point will cause a catastrophic failure throughout the Russian army to force a general withdrawal.