Is a sick woman obliged to wash impurities from her clothes for prayer if they are discharged involuntarily due to hemorrhoids, given the difficulty of washing due to severe cold?

The mother is not required to perform ghusl (ritual bath) upon seeing impurity. Rather, she is required to perform istinja' (washing away impurity) and remove the impurity, then perform wudu' (ablution) for prayer. If the impurity is continuously discharged, she is in the same position as a person with urinary incontinence (sahib al-salas), so she should perform wudu' after the time for prayer begins and pray with this wudu' both obligatory and supererogatory prayers until that time expires. She should also bandage the area unless it causes her harm.