In this paper, the Hermite-Hadamard inequality and its generalization for quasi $p$-convex functions are provided. Also several new inequalities are established for the functions whose first derivative in absolute value is quasi $p$-convex, which states some bounds for sides of the Hermite-Hadamard inequalities. In the context of the applications of results, we presented some relations involving special means and some inequalities for special functions including digamma function and Fresnel integral for sinus. In addiditon, an upper bound for error in numerical integration of quasi p-convex functions via composite trapezoid rule is given.