Is there a specific supplication from the Prophet, peace and blessings be upon him, to be recited when taking medication that has harmful side effects, out of fear of suffering from these effects?

There is no specific supplication to be recited when taking medicine. However, there are general remembrances (dhikr) and seeking refuge (ta'awwudhat) such as saying "Bismillah (In the Name of Allah)" or "Bismillahil-ladhi la yadurru ma' ismihi shay'un fil-ardi wa la fis-sama'i wa Huwas-Sami'ul-'Alim (In the Name of Allah, with Whose Name nothing on earth or in the heavens can cause harm, and He is the All-Hearing, the All-Knowing)" three times.

It is forbidden to take medicine if its harm is certain, unless the desired outcome from it is greater than the harm. In such a case, the lesser evil is committed to ward off the greater. However, if the two evils are equal, then taking it is forbidden. The assessment of harm is referred to the specialized physician.