They won't hit a "random target", they'll hit things like airfields, command centers, and ammunition dumps.

For example, a strike could take out several of the jets the terrorists in the Kremlin are using to kill civilians in Kharkiv.

Long range attacks are about knocking out supplies, the ability to produce new supplies, and the ability to get supplies to the front line.

Say Russia is getting more newly made artillery shells to the front line. The best way to fix this is to blow up the factories.

If you can't take out the factories directly, take out components that the factories need to operate: ore processing, fuel refineries, electricity grid, etc.

The problem is all the critical targets are a long ways from the front line and Ukraine currently has limited capacity to hit them.

Meanwhile Russia is targeting all of Ukraine's internal infrastructure constantly.

Long wars are won by the production capabilities of the groups involved as much as the front line troops.

If they were random targets, sure. But the messaging clearly included the wish to strike military targets at long range. Don’t have to lose people to a aerial bomb if the plane carrying it has been destroyed by a long range strike.

It slowed the progress by necessitating the movement of invading troops back to defend. Ukraine also had a major propaganda win from the Kursk offensive.

What benefit does striking some random target hundreds of miles within Russia accomplish

They’re not random but attempts to make more strategic difference, and to expand the war beyond just the front

• how can Russian artillery keep shelling if their supplies are blown up, and the supplies for those? And how responsive can they be at re-supply if new supplies have to come from hundreds of miles?
• how can Russia keep feeding the meat grinder if fresh troops need to fight their way to the front, lose their supplies, and take losses even before they get there?
• how can Russian commanders work if they’re dead? And their commanders are dead? And someone is trying to make battle decisions from hundreds of miles away?

Think of the Russian Black Sea fleet. The surviving ships are so far away that they’re not making any contribution to the war. Now, imagine making the Russian Air Force ineffective, Russian Command ineffective, and the supply situation ever worse

What benefit does striking some random target hundreds of miles within Russia accomplish for Ukraine, who is slowly losing the ability to hold ground?

The goal of the Ukraine conflict is to create a Pyrrhic victory for Russia so costly that it collapses from within. Long range missiles capable of destroying key domestic infrastructure and material reserves will (hypothetically) hasten that economic collapse.

If you can start hitting major financial centers, administrative offices, and social meeting spaces, you might even be able to decapitate some of the senior leadership and kick off a succession crisis in Putin's government. Alternatively, you push these leaders so far down into hiding that they can no longer effectively govern the country and foreign-sponsored insurgents can take over.

That's the theory of the war, at least. In practice, NATO leadership is still unsettled by the prospect of a Total War with a nuclear armed state. So they're not quite ready to drop a long range cruise missile on the Frunzenskaya Embankment, for fear of what a retaliatory strike would look like.

Everyone talked about tanks as ww1 winners. But people don't quite realise the stupendous artillery advantage the allies had. 2 or 3 to 1 by the end of the war in places. And significantly more shells. There's a reason we are still digging then up today