It's just a small utility subway
The tunnel involved in this hiatus appears to be neither that of the old Kingsway telephone exchange, which is located underneath Chancery Lane tube station, nor the Kingsway tramway tunnel, which runs under the middle of the road. It looks as though it's one of the pipe subways built by the London County Council in a few streets between 1861 and 1933 to carry gas and water pipes. Since then other utilities' pipes and cables have been added. AIUI they're big enough to walk through, but only just, and are at shallow with about 3 feet of cover.