In the article a partial integro-differential equation with degenerate kernel and Hilfer fractional operator is considered. The issues of unique classical solvability and construction of a solution to a mixed problem for a homogeneous integro-differential equation containing a Hilfer fractional analogue of the Barenblatt–Zheltov–Kochina operator are studied. The Fourier series method based on the separation of variables was used. By the aid of the Mittag–Leffler function is obtained a countable system. Sufficient coefficient conditions for unique classical solvability of the mixed problem are established. The absolute and uniform convergence of the obtained series is proved.