During the past 15 years, evidence for an ice-rich planet Mars has rapidly mounted, become increasingly varied in terms of types of deposits and types of observational data, and has become more widespread across the surface. The mid-latitudes of Mars, especially Utopia Planitia, show many types of interesting landforms similar to those in periglacial landscapes on Earth that suggest the presence of ice-rich permafrost. These include thermal contraction polygonal networks, scalloped terrains similar to thermokarst pits, debris flows, small mounds like pingos and rock glaciers. Here, we address questions concerning the influence of meltwater in the Utopia Planitia (UP) landscape using analogs of near-surface melting and drainage along ice-wedge troughs on Bylot Island, northern Canada. In Utopia Planitia, based on the identification of sinuous channel-like pits within polygonal networks, we suggest that episodic underground melting was possible under severe periglacial climate conditions. In UP, the collapse pattern and morphology of unconnected sinuous elongated pits that follow the polygon crack are similar to underground melting in Bylot Island (Nunavut, Canada). Based on this terrestrial analogue, we develop a thermal model that consists of a thick insulating dusty layer over ice-saturated dust during a period of slight climatic warming relative to today's climate. In the model, the melting point is reached at depths down to 150 m. We suggest that small-scale melting could have occurred below ground within ground-ice polygonal fractures and pooled in underground cavities. Then the water may have been released episodically causing mechanical erosion as well as undermining and collapse. After melting, the dry surface dusty layer might have been blown away, thus exposing the degraded terrain of the substrate layer.