Is money distributed to hospital staff at the end of the year considered unlawful if part of it comes from bank interest?

The fact that the hospital deposits its money in a Riba-based bank and earns interest on it does not prevent you from taking your due right from its funds, whether that is a bonus or a salary. This is because dealing with mixed funds is permissible when the money is mixed, and currency is not earmarked. Therefore, these funds are not forbidden for you, and you do not sin by benefiting from them. However, the sin lies with the usurer.