Hayate Gekkō was a special jōnin of the Hidden Leaf Village in Naruto. That is supposed to mean he's one of the village's top-ranking ninja, along with Kakashi Hatake and Might Guy. However, nobody would assume that simply from looking at him.
Hayate may have been an elite ninja, but he was also apparently rather sickly. He was constantly seen with dark rings under his eyes and a cough. Dying in his first battle of the series doesn't help his case either. With this in mind, one can't help but wonder just what his sickness was and how he still qualified to be a ninja.
Unfortunately, there's relatively little to say about Hayate. He proctored the Chūnin Exams, spied on Kabuto and Baki, got caught, died, and was reanimated for the Fourth Great Ninja War. He also apparently had a lover in Yūgao Uzuki. That's the extent of his role throughout Naruto. The reason for his sickness is never elaborated on.
The real point of focus is the Chūnin Exams, as it's where he's seen the most. When he first appeared to judge the preliminaries, all of the genin seem to be of the same mindset: This guy doesn't look like ninja material. In addition to the rings and the coughing, he was also struggling to find the words to explain the next phase of the exams. His cheekbones were also visible, which might indicate malnutrition. Whatever his condition, it seemed unlikely he was on par with the other jōnin.
It's possible Hayate had his apparent ailment since childhood. Back when he himself was taking the Chūnin Exam, he still had the rings under his eyes. While it's uncertain whether the rings are connected to his cough, it seems likely.
If Hayate really had been sick since he was young, then there are a couple of possible explanations for his cough. It's likely he had a disease that was slowly ravaging his body over the years. A less-morbid possibility is that he had trouble taking good care of himself; he couldn't get enough sleep and he didn't practice good hygiene. If it's the latter option, then perhaps he just had a cold or fever. It could either be something really major or really minor.
Regardless of Hayate's condition, he still seemed to be a proficient ninja. He was among the jōnin to stop Neji from killing Hinata, meaning his response time was just fine. He also still went out on stealth missions and could even engage in combat if need be. He was even the one who helped Yugao learn swordsmanship after she'd become a member of the Anbu Black Ops. Even if his affliction was a regular thing, it was far from holding him back.
