Is it possible to be kicked from Ausbildung because of Nadelstich verletzung?

[errant quarry]

You can not be kicked out for being "too much work". As long as you're not doing things like this purposely AND
As long as there's no reason to believe you're not capable of doing the work appropriately (eg you have very shaky hands and constantly get into things like this and there's no way to fix this issue), you're fine

Simply being clumsy is not a reason to be considered "incapable" of certain jobs. Bigger underlying issues might be

Tldr if anything, it's ur company issue for not teaching you the proper safety measures to take, you're still learning

[slender marlin]

I just looked away for a second while trying to deal with the needle and that was kinda my fault but nah I dont have any issues thank God, I was just worried since I had to go through so much procedure, D arzt and betriebs arzt and verband buch, so I got worried

[errant quarry]

Nah that's usual procedure