Is it permissible for me to refuse my mother's request for an additional, unagreed-upon sum of money, or to ask that it be accounted for within the loan between us, knowing that she does not immediately need it but rather for savings?

If the mother is not in need of the money she requests, you are not obligated to give it to her, and you will not sin by withholding it. There is no harm in considering what you give her as part of the loan installments. However, it is better not to deny her anything that would not harm you, because the mother's right is great, and being dutiful to her is among the best deeds.