Is it permissible for a husband to spend the night at his first wife's house, claiming his children are with her, given that he only grants his second wife one night?

The wife's right to qasm (spending the night with her husband) depends on the reason for her travel:

1. If she travels with his permission and for his need: Her right to qasm is not forfeited, and the husband must make up for the nights she missed upon her return. 2. If she travels without his permission or for her own need: Her right to qasm is forfeited.

If her right to qasm is forfeited, the husband is not obligated to spend every night with her. Instead, he is obligated to spend one night with her every four nights, just like a man married to one wife.