Can I spend my mom’s money for a new roof on her house?

Who is living in mom's house? A roof should have been done before the move to nursing home.

Is your mom on Medicaid? Or in the application phase?
If so why not sell the house and use the $$ for her care then if necessary apply for Medicaid.
It does not make sense to me to pay for a roof or put a down payment on one if no one is living in the house. If you (and your family) are living in the house then you should be the one to pay for the roof. If no one is there then sell the house and the cost of the roof would probably be deducted from the sale price unless the realtor says replacing the roof would actually increase the value and it would be recouped. Then it makes sense to put a new roof on.
And if you only put a down payment on the roof...who pays it off? Someone will at some point.

Why would you put a down payment on a new roof? Who is going to pay it off? If your mom has to spend down money for Medicaid eligibility, once her money is spent down, she will not have any money to pay ANY of her debts, she won’t be able to pay for the upkeep of the house either.

I don't think you can spend your mom's money on upkeep of her house when she is not in it. At least not if she is to have medicaid pay for her nursing home in the future. But others here know more about this topic.