Is it necessary to make up the day for one who swallowed phlegm residue after it reached the mouth area, due to the inability to expel it?

The questioner's fast is valid, because swallowing phlegm mixed with food that one was unable to expel does not invalidate the fast. A fasting person does not break their fast by swallowing food or drink that has not reached a place from which it can be expelled. If it was possible to expel it, and then the person deliberately swallowed it, then their fast is broken by that. As for what reached the letter Kha (خ) or Ha (ح) exit points, if the fasting person deliberately swallowed it, then their fast is broken by that. What the questioner mentioned about her inability to expel the rest contradicts its reaching the Ha (ح) exit point, because what reaches the apparent limit is not difficult to expel.