In the day of that song, the screw auger had not yet been invented, and it would be impossible to get enough leverage to drill a hole with a "pod auger"under water.

Now, if had had a screw auger (the type with the worm or screw at the tip, plus plenty of time, and he was drilling holes in a ship that was sheathed inside, theoretically, he could do enough damage if he did not penetrate the sheathing. That would let the water in, and it would run into the bilges behind the inner wall and would be very hard to locate the holes.

I believe that "Fothering" was developed in the 18th century, 200 years later.

Opinion, not possible.