What is undercutting?
Undercutting means digging deeper than the planned subgrade to remove unsuitable material, soft, wet, organic, or otherwise unstable soil, and replacing it with compactable fill or stone so what's built above doesn't settle.
You reach subgrade and the bottom pumps under the machine, or the soil is black with organics, or yesterday's rain turned it to pudding. Building on it anyway buries the problem under the new work. Undercutting removes it: dig until you reach competent material, then bring the grade back up with select fill or stone, compacted in lifts.
Getting paid for it
Undercut is the classic unknown, so treat it like rock: exclude it from the base bid, carry a unit price per yard of undercut-and-replace, and require notice and documentation when it happens. An inspector or geotech usually decides how deep is deep enough on spec work; on private work, photograph the proof, the rutted subgrade, the water, the organics, before you fix it.
Stabilization alternatives
Deep or wet undercuts sometimes get geotextile fabric or geogrid at the bottom before stone, which bridges soft spots with less depth. On the right soils, that fabric saves real yards of replacement material, and the supplier's spec sheet will tell you when it applies.