The main objective of this paper is to introduce and investigate a new class of preinvex functions, called approximately h-preinvex functions, which depend on a given bifunction ε(·, ·). It is observed that this class includes several other new and known classes of preinvexity. We derive a new refinement of Hermite–Hadamard-like inequalities using the class of approximately preinvex functions. We also derive a new q-integral identity using the q-differentiable function. Using this auxiliary result, we obtain several new estimates of quantum bounds. We also discuss several new special cases of the obtained results. Certain applications of the main results are also discussed.