Refugees attempting to build the blockhouse on the remains of the fort never mounted ... cannons although they may have had at least one with them.
The British needed a stable, local supply of wood and that was just across the river in Bergen County. Ships were broken up for firewood and it was extremely difficult to get wood from Long Island.
It's paper archaeology from the pension applications, the newspaper accounts, the correspondence of George Washington, and some British journals.
The Loyal Refugee Volunteers were not members of the regular British Army. Their job was to cut wood for the British, who would pay them for it. To supplement that wood income, they raided into the Bergen County countryside to take horses and cattle, which they then would sell.