My daughter has a 2013 Elantra with the 4cyl engine.
When pulling away from a stop or applying the brakes to stop there is a Single loud click from the front of the car(sounds like below the car too).
It seems too high pitched to be a motor mount.
Car has 140,000 miles on it.
Might also be worn caliper slides, or possibly a CV joint? But more likely the above suspension pieces.
Do you get the click if you sit still and shift back and forth between reverse and drive? If so, that points to a CV joint, if not, to one of the other possibilities. Take it to your mechanic to check it out.
The symptoms are consistent w/some sort of front suspension system play. Experienced shops are presented with this sort of problem all the time, so know how to diagnosis the cause. Suggest to not defer a shop inspection, as play in the front suspension system could present a safety hazard. If car is equipped w/front struts, a problematic strut mount would be my first guess.