What gotra do we say in the saṅkalpa if we don’t know it, or for an adopted or interfaith child?

an adopted child takes the adoptive father’s gotra; when no gotra is known, priests commonly use Kāśyapa gotra.

The gotra is the traditional name of the sage-lineage a family belongs to, and it is recited in the saṅkalpa of most pūjās. An adopted child is traditionally brought into the adoptive family’s gotra. For a child of an interfaith marriage, the Hindu parent’s family gotra is generally used for Hindu rites.

Many families, however, have simply lost track of their gotra over generations, and people who come to Hinduism later have none. In such cases a widely followed custom is to use Kāśyapa gotra, since sage Kaśyapa is regarded in the Purāṇas as a progenitor of many beings, so everyone can be considered his descendant. Some priests instead say “Śiva gotra” or “Viṣṇu gotra”, following regional practice.

No pūjā fails for lack of a gotra. Saying your name with devotion is enough. If your situation is specific, Ask Chiluka.

Source: common priestly practice — Kāśyapa gotra when unknown

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