Is money earned from working in a private clinic during official working hours at a public hospital permissible or impermissible (halal or haram)? And does the ruling differ if the work in the private clinic is after the official working hours at the public hospital?

It is not permissible for an employee to work for anyone other than their employer during official working hours except with their permission, because they are a private employee whose benefit is estimated by time, and the employer owns their benefit during this period. If they do so, the employer is entitled to the value of the benefit that was missed or what the employee took from working for someone else. As for working outside of official working hours, there is no objection to it.