A version of a Turing maching that can cut out and/or remove tape squares

Question

I would like to know if in the literature there have been considered versions of Turing machines that, instead of changing the content of one tape square at a time are allowed to perform more general moves, particularly adding new squares to the tape and cutting out existing squares. Specifically, where in a single transition the head can also insert new tape squares and or remove some new tape squares, say to the right and or to the left of the current position of the head, write something on any newly inserted tape squares, change the letter on the square that was being read, and then move left, right or stay put. Or some version of the above. If some such versions of Turing machines have been considered, and if there are some standard names/terminology for them, I would very much like to know what they are. Thanks!