Is it permissible for my mother, who suffers from a chronic illness and is unable to fast Ramadan, to pay the expiatory fidyah for not fasting to one person all at once a long time after Ramadan, or to pay it in installments?

If your mother's illness is chronic, with no hope of recovery, and she cannot fast because of it, then she must feed a poor person for each day. According to the majority of scholars, giving money in place of feeding is not permissible. Feeding is obligatory for the able-bodied, to be performed immediately. It is permissible to give the feeding to a single poor person for each day the sick person missed. The feeding can be done in one of two ways: either by preparing food and inviting the poor to it, according to the number of days owed, or by giving them uncooked food.