1
I work long hours, so I don’t stay with my children very much.
I work long hours, so I don’t spend much time with my children.
I work long hours, so I can’t be with my children very much.
Use be with someone or spend time with someone when you mean that you are together: ‘I enjoy spending time with my family.’
Stay with someone normally means to live in that person’s home temporarily: ‘We’re staying with some friends while our house is being repaired.’
In the red sentence, we are talking about the amount of time the parent and children spend together, not about living temporarily in the children’s home.
2
We’re going to stay at Uncle Peter for a few days.
We’re going to stay with Uncle Peter for a few days.
We’re going to stay at Uncle Peter’s for a few days.
Use stay with + person: ‘We’re staying with some friends.’
Use stay at + possessive to mean someone’s home: ‘We’re staying at my aunt’s.’