We must await the full archaeology report but it is clearly a medieval or post-medieval corn-drying kiln, very simillar to that built into the conserved section of Hadrian's Wall, slightly further east, across Towne Gate.
The excavation was undertaken by an archaeologist from Wardell Armstrong, and my photos taken, with careful regard for health & safety, through the wire fence at the east entrance to the building site.
The location of the kiln is likley to lie close to, but just to the north of the line of Hadrian's Wall, which hadn't been visible in this area of the site.
Behind the circular part of the kiln in the foreground is a deep fire-pit bounded by large masonry on the south side.
Wardell Armstrong - Tulip Mews, Heddon-on-the-Wall, Northumberland. Watching Brief Report (2020).