I came home late when my child was sick
Why did you come home late?
The only reason is my job.
Sounds like you have a busy job, which is understandable, but if you knew your child was sick, you should have worked something out with your job beforehand. Your child needs care and you're one of the only people who can provide it.
I get that but I'm the only one working in the house. My wife doesn't work, and my kid is old enough to take care of themselves.
Seeing as you wife was available, you being late from work while you child is sick shouldn't be too big of an issue. Your wife is there to take care of them so you shouldn't need to feel guilty about it.