While undertaking kainkaryam of Tirumanjanam at temples or during Sudarshana homam at homes, can the sankalpam by prapannas include “bhagavad prItyartham”? Or is it appropriate to do the sankalpam with one’s gotram, nakshatram, name, etc.?

Prapannas sankalpam must include “bhagavad prItyartham”. At the same time, for certain sankalpams, gotram, nakshatram might need to be invoked. Even as per the scriptures, certain rituals might require sankalpam to include gotram, nakshatram, etc.
Similarly, for Tirumanjana kainkaryam at temples, one can include gotram, nakshatram, name in sankalpam, or can choose not to do so. Instead, they can invoke ‘bhagavad prItyartham’.
It appears it is a practice to include name, gotram and nakshatram in the sankalpam for Sudarshana homam performed at home as they are performed for the health and wealth benefits of the household members. Even here, one can include ‘bhagavad prItyartham’ followed by name, gotram and nakshatram in the sankalpam.