Marvel's Weapon H might be the product of Wolverine and the Hulk's DNA, but one of his strangest weaknesses and powers is a much weirdest version of Logan's adamantium claws. In Hulkverines #1 by Marvel Comics, Clayton Cortez accidentally pops his claws from his body after a sneezing fit. It turns out that Leader was able to infect Weapon H with a gamma-virus that altered his healing factor, leading to his claws appearing when the hero didn't mean to bring them out.
Marvel's Weapon H is a combination hero who Weapon X made in their efforts to combine the Wolverine and the Hulk. Mercenary Clayton Cortez was changed into the hybrid hero, as he could Hulk out and had Wolverine's adamantium-coated claws. Weapon H also combined several other characters as Lady Deathstrike, Domino, Sabretooth, and Warpath's DNA was also used in the experiment to create the monster, giving him a wide array of powers that are still being revealed to this day in the pages of Marvel Comics. However, Marvel Comics once showed Weapon H has a weakness Wolverine never had that bizarrely impacts his powers.
In Hulkverines #1 by Greg Pak, Ario Anindito, Morry Hollowell, Andrew Crossley, and VC's Joe Caramagna, Clayton Cortez tries to hide out with his family as the Hulk and Wolverine try to track him. However, unbeknownst to Cortez, Leader has infected him with a gamma-altering virus, which means when he sneezes, his adamantium claws unsheathed. The virus hits Weapon H's healing factor, as he's not supposed to be able to catch a cold, as the villain exploited a weakness related to his Hulk side.
Based on his cold and claws unsheathing when he sneezed, Clayton Cortez realized that he needed to get his family far away from him as he couldn't control his powers due to the weakness from Leader's gamma-virus. When Bruce Banner later confronts Weapon H, he continues to sneeze as his claws pop out of his body. While Weapon H's adamantium claws work differently than Wolverine's, as the nano-tech in his body allows him to shoot them like projectiles, the sneezing weakness is an exploit that wouldn't work on Logan. By infecting his Hulk powers with a gamma-virus, Weapon H's weakness was attacked, as his continued sneezing meant he couldn't control when his claws popped out - making him dangerous to his own family.
Ultimately, the gamma-virus proved to be a smart way to attack Clayton Cortez and one of his biggest weaknesses. Since he was part Hulk just as much Wolverine, Leader was able to infect that side of him and alter his healing factor, giving him a cold that led to his claws being out of control when he sneezed. It might seem like a silly weakness for Weapon H, but it was a genius way to attack the Hulk-Wolverine hybrid hero.