The main goal of this article is to present multiple term refinements of the well-known Jensen's inequality for $h$-convex functions for a non-negative super-multiplicative and super-additive function $h$. For example, we show that \begin{align*} &h(1-v) f(0)+h(v) f(1) \geqslant f(v) +um_{n=0}^{N-1} h(2r_{n}(v)) um_{k=1}^{2^{n}} \Delta_{f,h}^{(0,1)}(n,k) \chi_{eft(\frac{k-1}{2 ^n}, \frac{k}{2^n}\right)}(v), \end{align*} for the $h$-convex function $f$ and certain positive summands. The significance of the obtained results is the way they extend known results from the setting of convex functions to other classes of functions.