When is the last time time the New England Patriots scored a touchdown on a punt return?

The onside kick from scrimmage was eliminated, in the collegiate game, before WWI -- around 1912, I believe. The NFL started in 1920. So I guess the answer is -- never. In the NFL, an onside kick is only possible on a kickoff or on a free kick after a safety. But has there ever been an onside drop kick? I don't know, but lets consider why that would rarely (if ever) happen:

Kickoffs are required to be a place kick (from a tee). So the only time you could even attempt an onside drop kick is after a safety, which is one of the rarest plays in football.

An onside kick after a safety is very dangerous, as the kick must be from the 20 yard line. The opponent could recover the ball already in field goal range.

An onside kick must hit the ground to prevent the other team from calling for a fair catch. This is more difficult to pull off with a drop kick.