In this paper, we introduce a new class of partially degenerate Hermite-Bernoulli polynomials of the first kind and generalized Gould-Hopper-partially degenerate Bernoulli polynomials of the first kind and present some properties and identities of these polynomials. A new class of polynomials generalizing different classes of Hermite polynomials such as the real Gould-Hopper, as well as the 1-d and 2-d holomorphic, ternary and polyanalytic complex Hermite polynomials and their relationship to the partially degenerate Hermite-Bernoulli polynomials of the first kind are also discussed.