Euclidean Clifford analysis has become a well-established theory of monogenic functions in higher-dimensional Euclidean space with a variety of applications both inside and outside of mathematics. Noncommutativity of the geometric product in Clifford algebras leads to what are now known as inframonogenic functions, which are characterized by certain elliptic system associated to the orthogonal Dirac operator in (Formula presented.). The main question we shall be concerned with is whether or not a higher-order Lipschitz function on the boundary Γ of a Jordan domain (Formula presented.) can be decomposed into a sum of the two boundary values of a sectionally inframonogenic function with jump across Γ. To this end, a kind of Cauchy-type integral and singular integral operator, very specific to the inframonogenic setting, are widely used.