The dynamics of a system defined by an endomorphism is essentially different from that of a system defined by a diffeomorphism due to interaction of invariant objects with the so-called critical locus. A planar endomorphism typically folds the phase space along curves <em>J</em>₀ where the Jacobian of the map is zero. The critical locus is the image of <em>J</em>₀, denoted <em>J</em>₁, which is often only piecewise smooth due to the presence of cusp points that are persistent under perturbation. We investigate what happens when the stable set <em>W</em>^s of a fixed point or periodic orbit interacts with <em>J</em>₁ near such a cusp point <em>C</em>₁. Our approach is in the spirit of bifurcation theory and we classify the different unfoldings of the codimension-two singularity where the curve <em>W</em>^s is tangent to <em>J</em>₁ exactly at <em>C</em>₁. The analysis uses a local normal-form setup that identifies the possible local phase portraits. These local phase portraits give rise to different global manifestations of the behaviour organised by five different global bifurcation diagrams.