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This manuscript aims to study a generalized, set-valued, mixed-ordered, variational inclusion problem involving <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">H</mi><mo>(</mo><mo>·</mo><mo>,</mo><mo>·</mo><mo>)</mo></mrow></semantics></math></inline-formula>-compression XOR-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>α</mi><mi mathvariant="script">M</mi></msub></semantics></math></inline-formula>-non-ordinary difference mapping and relaxed cocoercive mapping in real-ordered Hilbert spaces. The resolvent operator associated with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">H</mi><mo>(</mo><mo>·</mo><mo>,</mo><mo>·</mo><mo>)</mo></mrow></semantics></math></inline-formula>-compression XOR-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>α</mi><mi mathvariant="script">M</mi></msub></semantics></math></inline-formula>-non-ordinary difference mapping is defined, and some of its characteristics are discussed. We prove existence and uniqueness results for the considered generalized, set-valued, mixed-ordered, variational inclusion problem. Further, we put forward a three-step iterative algorithm using a ⊕ operator, and analyze the convergence of the suggested iterative algorithm under some mild assumptions. Finally, we reconfirm the existence and convergence results by an illustrative numerical example.