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Can it leave other bits behind after it moves?

By definition of a pure spaceship / glider, no it can't. If it emits persistent "exhaust" that's a "smoking ship"[0].

IIUC, "smoke" is ephemeral by definition. A spaceship/glider that leaves persistent "debris" in its wake is called a "puffer":

Smoke refers to fairly long lived debris that eventually dies out, the term for one that leaves permanent debris is a puffer.

No, if it does it's called a puffer.