Here are two of the more famous team-ups of the Man of Steel and the Manhunter from Mars, where it seemed to more of battle than a social!
World's Finest Comics #212
That problem was on the planet Vonn, where J'onn's Martians had settled after leaving Earth (and where the Martian Manhunter had followed them to). The Martians were captured by the Thythen, who were using their lifeforce to charge their Robo-Chargers. The Martian Manhunter ended up in an ancient alien stronghold, that teleported to Earth. All this stress unhinged J'onn, so he attacked Superman, but the two transported back to Vonn with the structure, a planet under a red sun, so Superman was powerless. The pair went off to fight the invading aliens, meeting J'onn's female friend and fellow Martian, Bel Juz. Superman worked to get her alone, suspecting her of being a traitor, while the Martian Manhunter faced the Thythen, with the help of the Robo-Chargers (the Martian lifeforce powering the robots allowed the Martians to control the robots, which would also help them rebuild this world).
move to the Motor City!).
DC Comics Presents #27
Superman was enlisted by interplanetary thief, Mongul (by kidnapping Lois Lane, Jimmy Olsen and Steve Lombard), to retrieve the Crystal Key that controlled Warworld, which was being protected by the Martian Manhunter (who had gotten the key from the Largas, who were an ultra-peaceful race that came upon the inactive Warworld). A super battle between Superman and the Martian Manhunter ensued, with Superman getting the Key, but having to give it to Mongul to free his friends, and even with Martian Manhunter's help, Mongul escaped with the Crystal Key, proving Superman fallible, much to the anger of the Martian Manhunter.
battle with Mongul continued on after this alien's first appearance, with Superman getting assistance from others as well, like his cousin Supergirl and an alien Starman.