Your custodial parent is your father because you live with him. It’s his income that goes on FAFSA. Money that can be attributable as “gifts” to him are not considered income. Anything that you can attribute as going to him as a gift that is then used to run the household should not be reported as income. If your mother clearly has it recorded and indicated that the $10k she gave your father in 2018 was child support, then it has to be listed as such. If it’s just a gift to him or shows as such, then that’s a gift.
Income for FAFSA purposes and financial aid are totally different things. However, monetary gifts to your father are treated pretty much the same; the IRS does not charge taxes on monetary gifts, and FAFSA does not expect a parent to report monetary gifts either.