Respect threads are just plain easier to make than capability threads. Hence this should not be regarded as a definitive representation of Cyborg Superman's limits and capabilities; it's more like Hank's greatest hits.
Adventures of Superman #468: This is from before he got the half-Superman/half-Terminator body. Henshaw is capable of emitting energy pulses that could shut down all forms of electronics all over the east coast.
Superman #78: Steals Doomsday's body from a military installation, flies into space, calculates a trajectory that will ensure that his body floats through the cosmos forever, and then chucks Doomsday into space and (it is later revealed) out of the solar system.
Superman #79: Accesses almost every computer system in the world via the White House computers.
Adventures of Superman #503: Beats the living daylights out of Superboy.
Superman #80: Fights Eradicator and dominates fairly easily until Mongul blows up Coast City in the biggest explosion ever seen on the planet. The magnitude of the blast was said to surpass any weapon invented by man, which would presumably include nukes. The explosion leaves Eradicator barely alive but Henshaw is virtually unscathed.
Superman #81: Makes Mongul his b!tch.
Superman #81: Again....
Adventures of Superman #504: .....and again.
Superman #82: Fights Superman, Steel, Superboy, Supergirl and Eradicator. He takes control of Steel's metal body to make him strangle himself and then fights Eradicator and a depowered Superman. Kryptonite enters the picture and everyone starts to get weaker, until the Eradicator sacrifices himself to restore Superman's powers completely. By that point Superman was good as new and Henshaw was weakened from the Kryptonite, so there you go.
Superman/Doomsday: Hunter/Prey #1: Creates a body out of an Apokoliptian soldier's corpse, and that body is powerful enough to injure Superman to a notable degree (Supes then gets up and flies away like nothing happened, but that's more than likely because of the Mother Box on his arm, it heals it's user).
Superman/Doomsday: Hunter/Prey #2: Takes control of the Apokoliptian communications system and uses it to murder the entire army.
Superman/Doomsday: Hunter/Prey #2: Fights Superman who gets help from Darkseid. Henshaw survives a hit from the Omega Beams (Darkseid was still recuperating at the time, though) and then is destroyed by a second hit of the OB. (He was actually transmuted, but whatever)
Superman #107: Stomps Superboy, Supergirl, Steel and Eradicator all at the same time.
Superman #108: Tribunal-Prime takes control of Henshaw's body for a while and uses it to one-shot Superman.
Superman: The Man of Steel #52: Takes control of the entire tribunal planet with his technopathy.
Adventures of Superman #531: Teleported by the tribunal to the event horizon of a black hole, it sucks him in although ultimately not damaging him to any visible extent (it's part of the plot to send him somewhere).....
Green Lantern/Silver Surfer: Unholy Alliances: .......and he comes out unscathed in the Marvel universe, where the first thing he does is destroy a planet. The Silver Surfer fights him and they stalemate for a while until Parallax shows up.
As far as I know, this crossover is very much canon since it's a part of Henshaw's story (it's not unlike when Invincible entered a portal to Marvel and met Spider-Man, and that crossover is canon for the same reason).
Confirmation that the story is canon: Hal talks about how he pursued Henshaw into another universe in Parallax: Emerald Night.
Parallax: Emerald Night #1: Fights Parallax. Heals himself in seconds when Hal blows a hole in his chest. Hal then merges him with the Source Wall, imprisoning him there.
Superman #128: But that doesn't stop him, he takes control of the portion of the Wall that he's imprisoned in and uses it to pull Superman into it. Considering that the Source Wall is a very one-of-a-kind object in the DCU, it speaks volumes about Henshaw's ability to control and animate all kinds of things.
Adventures of Superman #551: Creates a small army of dupes from portions of the Source Wall to fight Superman. At the end of the fight Superman thinks he's beaten Henshaw and imprisoned him in the wall again, but it was a ruse; Henshaw had bonded with Superman's energy suit to hitch a ride back to Earth without his knowledge.
Superman #129: Superman tries to restrain Henshaw in an energy grid, but he escapes by bonding with a small model of himself.
Superman Red/Superman Blue: The model falls into the hands of Toyman, and Henshaw then leaves it to create a new body for himself out of some nearby children's toys (and even that body is capable of fighting Superman).
Superman Red/Superman Blue: With a body formed out of children's toys, Henshaw fights Superman. Supes tries to restrain him in an energy field, Henshaw builds a device on the spot to break free. He then adjusts his heat vision so that it can hurt even an energy being, and then forces Superman into a position where he had no choice but to bear the brunt of Henshaw's heat vision for a while. Note that Superman himself said he couldn't stand the heat, and remember that Henshaw was doing all this with a body formed out of children's toys. Then Toyman shows up and one-shots Superman with a gun that in all probability was created by Hank.
Adventures of Superman #563: Demonstrates that his powers do not wane in the absence of sunlight the way Superman's do, and some shape-shifting.
Superman: The Man of Steel #100: Escapes from the Phantom Zone and overpowers Superman with yet another new body until a Kandorian strike team shows up to restrain him.
Green Lantern #11: One-shots Hal Jordan (albeit by attacking him from the back) while in possession of several power rings.
Green Lantern #12: Easily defeats Guy Gardner.
Green Lantern #12: Uses his own form of mechanized mind-control, the willhunters, on Hal. For reference's sake, Henshaw had all the "lost Lanterns" strung up and docile because his willhunters had subdued their minds. Hal manages to break free of their control, but it's later revealed that he could do so because Arisia helped him.
Green Lantern #13: Endures an attack from Hal that was as hot as the sun.
Tales of the Sinestro Corps: Cyborg-Superman: Stomps Superman with several Qwardian rings.
Green Lantern #25: Tanks a blast from a Guardian, and is then hit along with the Manhunters and the Anti-Monitor with a galaxy-busting blast from the GLC. His body is destroyed but his consciousness survives and the Manhunters reboot him.
Green Lantern Corps #49: Takes control of the Alpha-Lanterns.
Green Lantern Corps #50: Manhandles Ganthet and drains his ring's power into a battery in his chest.
Green Lantern Corps #50: The planet Grenda was inhabited by over a billion robotic lifeforms. Henshaw enslaved them all with his technopathy.
Green Lantern Corps #51: Commands an Alpha-Lantern to kill himself, and the AL complies (I like to think this is what would happen in a fight between Henshaw and Beta Ray Bill).
Green Lantern Corps #51: Makes Ganthet cry. This feat is here simply because it amuses me.
Green Lantern Corps #52: The Green Lanterns on Grenda bring all of its one billion inhabitants to fight Henshaw, and while he is focused on pushing them back the GL's attack him. He leaves his body and hops into Boodikka's, where she fights him on some weird astral-like plane and "kills" him. Seems a bit ridiculous, but no matter, because......
Justice League of America #55: ...........he wasn't dead, he simply waited until the time was ripe and then tore himself out of Boodikka's body during Reign of Doomsday.
Superman/Batman Annual #5: Creates a new body within seconds from Boodikka's corpse and portions of the JLA Watchtower, and then takes control of the Watchtower with his technopathy.
Superman/Batman Annual #5: Fights ROD Doomsday. While the fight is going on, the Watchtower registers Henshaw and his presence as rogue Kryptonian activity and activates the default safety measure: a magic cocktail created by Zatanna and Dr. Fate. It weakens Supergirl severely but does not even slow down Henshaw. The actual fight itself goes back and forth for the most part until Henshaw works himself up into a rage at the prospect of defeat, and then blitzes Doomsday, tearing him clean in half. Which in itself is a pretty straightforward victory, but then Doomsday comes back to life and uses his plot powers to copy Henshaw's powers and......the rest is best left unsaid.