I just cocked and fired into the ground until I was out...
Forget about break-in. I'm not saying that break-in has no effect on spring/gas piston air rifles, but it doesn't take a gun that groups terribly and turn it into a really accurate shooter over the course of a few pellet tins. It will smooth out the firing cycle somewhat and that can help with accuracy.
What can improve terrible accuracy is ammo the gun likes, and good spring gun technique.
If I were in your situation, I would get some good quality lead pellets, at least a couple of different kinds and with a dome/round-head profile and an hourglass/diabolo body shape. Not sharp tips, no plastic or non-lead penetrator components, not lead-free, not copper-plated, no flat sides, not any of the "cool" pellet designs they sell these days. Something with a weight of about 14-15gr, not lighter, but maybe a little bit heavier if you want. I wouldn't go over 20gr. Something by JSB, H&N, or maybe RWS. Crosman if you don't want to order something online.
Then I would check all the stock screws on the gun and all the scope screws to make sure they're tight. I would look at the scope mounting hardware and scope to make sure none of it is showing marks that indicate anything is slipping or moving. Generally speaking, if I have a gun with iron sights, I try to shoot it some with iron sights before mounting an optic just so I have one variable eliminated before I add in the variables of the optic and optic mounting hardware.
Forget about cleaning the barrel. First of all, firearm solvents should never be used on airguns. Second, it's just not required. Powder/carbon fouling is non-existent, and leading is rarely an issue.
The gun is going to move in recoil--it's inevitable. What's different in a spring-piston gun is that the movement starts BEFORE the pellet starts moving. The piston that compresses the air for the shot is what generates the recoil. Pulling the trigger releases a piston that drives forward under spring pressure to compress the air in the compression chamber. Recoil begins the instant that the piston starts moving. When the compression gets high enough, the pellet finally starts moving. The piston keeps going forward until it either slams against the front of the chamber (not ideal) or is stopped by the now highly compressed air at the front of the chamber. Either way, that causes a significant forward recoil impulse--but that part of recoil likely happens after the pellet leaves the barrel. The important part of recoil is what is generated by the initial movement of the piston as it goes forward to compress the air.
Since the gun is moving (a lot) before the pellet even starts moving, how the gun is restrained will have a large effect on where the muzzle is when the pellet exits. And that, in turn, will have a large effect on where the pellet hits on the target.
You can't keep the gun from moving in recoil, so the alternative is to set it up so that the movement is extremely consistent. So nothing can touch the gun that would allow it to bounce or vibrate during recoil. Nothing hard. If shooting from a bag, you want the bag to be kind of slick so that when the fore-end slides on the bag, it can do so smoothly without catching or jerking. You want to be positioned the same for every shot and the way you grip the stock to pull the trigger needs to be consistent. The pressure of the gun against your shoulder needs to be consistent as well as the exact positioning of the stock on your shoulder and the tension in your shoulder muscles and your shoulder position relative to your body. A little bit of pressure on one side of the stock while you pull the trigger can throw the shot to one side or the other. A little bit of different shoulder tension can cause a significant vertical deflection of the shot.
Where your cheek touches the gun matters--how hard it's pressed against the stock matters. Where the stock rests on the bag matters. Some people put tape on their stocks to insure consistent positioning of their hands/cheek/bags when shooting.
For whatever it's worth, I don't benchrest my springers. It's a hassle, for one thing, and for another, they shoot to significantly different point of aim from the bench than when shot offhand. But I guess you need to prove to yourself that the gun can group well, and maybe shooting from the bench is a way to get that figured out.
The nice thing is that once you get your technique and consistency good enough to achieve acceptable results from a springer, shooting firearms seems much easier. You can be a lot sloppier with a firearm than with a springer and still get good results.