Is it permissible to request financial assistance from the state, without misrepresenting information, and with the father being able to provide?

It appears from the question that the inquiry about the income of someone living with you is regarding the sufficiency of their means. If the father provides for his daughter, but he does not live with her, then the condition for receiving assistance is not met. And if the father provides sufficient support for his daughter, then her request for assistance is not free from the blameworthiness mentioned in asking people for more than what is needed.