Could a neck puncture effect your speech?

Maybe: More details are needed but in theory injury to the recurrent laryngeal nerve from a puncture wound could lead to hoarseness. It would be uncommon as an isolated injury as there are many other important structures nearby that would also likely be injured.

Possible not probabl: While it is possible and could be due to nerve injury as mentioned by dr. Handler or to a variety of other theoretical injuries such as a hematoma, fibrosis, direct vocal cord injury, this is not probable.