What is the ruling on a woman breaking her fast in Ramadan, given that she suffers from extreme lethargy and fatigue that prevents her from completing the fast except by sleeping, and what is the expiation for one day?

If an illness is feared to worsen, delay recovery, or cause harm by fasting, then breaking the fast is permissible. If the illness is expected to be cured, then it is obligatory to make up the missed days when able to do so, and expiation does not suffice for making up the fast. It is permissible to delay making up the fast until after the following Ramadan (definition) if the excuse of illness persists, and in this case, no fidyah (definition) (ransom) is required.