What is the ruling pertaining to performing prayers in a mosque built using funds where the lawful and the prohibited are mixed, such as funds from usury or the income from unlawful/illegal activities?

In accordance to Islamic principles, performing prayers in that mosque is allowed and the sin of earning prohibited (haram) money returns to its owner.

Unless the mosque is wholly built using funds of persons who are involved in acquiring prohibitive (haram) money, and the intention of leaving prayer in the mosque that these persons are building is the disavowal of their actions and a means of enforcing them to get back the right path and as long as that it does not lead to relinquishment of Jamaah and Friday prayers.