What does people mean when they call Izuku a "problem child"?
Why is Izuku called a ‘problem child’?
Aizawa inaugurated the name in chapter 136.
Izuku is far from being a bad kid, but it’s true his strong drive to help and protect others at the expanse of his own well-being can be problematic. Izuku badly hurt himself multiple times because of that, so much he was close to lose functional arms after his fight against Muscular. Because of the adrenaline rushes, Izuku often forgets his pain and tends to hurt himself even more because he’s not even thinking about the state of his body anymore. Aizawa witnessed that more than once, particularly during the hellish night of the training camp.
Even if it’s illegal and dangerous for him, Izuku will rush into some situations he really shouldn’t butt in and can’t be stopped; that’s what Aizawa was admonishing him for. He saw Izuku being badly injured multiple times and must be frustrated, because he realized he couldn’t totally prevent it. Izuku, and all of the students have their own will and there are things you can’t stop them to do even if it’s for their own good. Izuku’s quite particular about it, because of his absolute urge to save. Even if he tries to stop him for his own sake, Izuku won’t listen and will find a way to do something anyway, like he did when he finally went to Kamino to save Bakugou with Kirishima and others. So it’s exhausting for Aizawa who has to cover for him each time. Izuku’s still young, so he might not realized all of the responsibilities Aizawa’s facing for his behavior, all of the paperwork and administration stuff he’s putting others into because of his reckless acts. Because of the Stain incident, Gran Torino saw his right to teach revoked for six months for example. As a teacher, Aizawa is the one responsible for those students, and they kinda make it hard for him to take a break from those responsibilities. To warn them, Aizawa stated he was close to expelling them, but also that they broke his trust and it would take time for it to be restored.
When there’s a problem, you can be sure Izuku’s there to be found.
So during chapter 136, after it was decided Izuku would also take part in Eri’s rescue, Aizawa reminded Izuku about those warnings when he called him a problem child.
It’s not that Aizawa’s fed up with his students. Aizawa cares about his students, and worries about them, so all those reckless things they do are as exhausting for him as they’re dangerous and deadly for his students. So him calling Izuku a ‘problem child’ while crouching so he can properly face him and fist-bumping him in the chest is more a sign of benevolence. He knows well Izuku. So if he can’t totally prevent Izuku from helping out, the last he can do is seriously warning him and keeping a close eye on him to keep him safe!