Is it necessary to recite the supplication for expiation of a sitting (kaffarat al-majlis) every time a person gets up from a gathering for a temporary reason, or is it sufficient to say it once at the end of the sitting? And is it a condition to say the supplication for expiation of a sitting only while actually sitting?

This remembrance is sunnah (recommended) and not obligatory. Whoever recites it will have their idle talk in the gathering expiated. We have not found any statements from the commentators regarding reciting it upon leaving a gathering with the intention of returning to it. However, if one recites it, it is good, and perhaps reciting it once at the conclusion of the gathering would suffice. Some scholars have mentioned that for someone who leaves the mosque with the intention of returning shortly, the "Tahiyyat al-Masjid" (greeting prayer for the mosque) is not sunnah for them, because their departure is not considered a complete leaving. Similarly, it can be said that rising from a gathering with the intention of returning shortly is not considered a complete leaving, and reciting the remembrance in this situation is a good thing.