Is it permissible to combine the intention of spending on one's family with the intention of expiation for not fasting Ramadan, given the inability to fast due to health reasons and the inability to pay the expiation separately?

It is not permissible to give expiation (kaffarah) to your mother because her maintenance is obligatory upon you. However, it is permissible to give it to a poor sister who does not possess enough to meet her basic needs. The amount of expiation according to the Hanbalis is half a Sa' of food (approximately 1.5 kg), and according to the Shafi'is and Malikis, it is one Mudd of food (approximately 750 grams) for each day. Thus, the expiation for a month would be 45 kg according to the Hanbalis, and half of this amount according to the Shafi'is and Malikis. Paying the monetary value is not sufficient according to the majority of scholars, but it is sufficient according to the Hanafis. All of this applies if the illness is not expected to be cured. However, if recovery is hoped for, then one must make up the missed fasts after recovery.