The white blood cells primarily responsible for adaptive immunity are

T and B cells.

Explanation:

Adaptive immunity is the kind of immunity that you acquire after exposure to an antigen (pathogen or vaccine), it involves memory in order to facilitate a quick response if there's a reinfection with the same pathogen. T cells are responsible for the cell-mediated response while the B cells are responsible for the humoral response. Both of these cells are specific to the pathogen's structure, therefore they can kill it directly.

I hope you find this information useful and interesting! Good luck!