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We initiate the study of m-subharmonic functions with respect to a semipositive (1, 1)-form in Euclidean domains, providing a significant element in understanding geodesics within the context of complex Hessian equations. Based on the foundational Perron envelope construction, we prove a decomposition of m-subharmonic solutions, and a general comparison principle that effectively manages singular Hessian measures. Additionally, we establish a rooftop equality and an analogue of the Kiselman minimum principle, which are crucial ingredients in establishing a criterion for geodesic connectivity among m-subharmonic functions, expressed in terms of their asymptotic envelopes.