The fuzzy order relation ≽ and fuzzy inclusion relation ⊇ are two different relations in fuzzy-interval calculus. Due to the importance of p-convexity, in this article we consider the introduced class of nonconvex fuzzy-interval-valued mappings known as p-convex fuzzy-interval-valued mappings (p-convex f-i-v-ms) through fuzzy order relation. With the support of a fuzzy generalized fractional operator, we establish a relationship between p-convex f-i-v-ms and Hermite–Hadamard (ℋ–ℋ) inequalities. Moreover, some related ℋ–ℋ inequalities are also derived by using fuzzy generalized fractional operators. Furthermore, we show that our conclusions cover a broad range of new and well-known inequalities for p-convex f-i-v-ms, as well as their variant forms as special instances. The theory proposed in this research is shown, with practical examples that demonstrate its usefulness. These findings and alternative methodologies may pave the way for future research in fuzzy optimization, modeling, and interval-valued mappings (i-v-m).