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.
Sometimes a worn sway bar bushing can cause that kind of sound.
Also sometimes the front lower control arm rearward bushing will degrade (blow out) and clunk when doing that…
Have someone watch the front wheels, from the side, to see if it shifts forward/rearward when stating to move and braking…
Bushing lower left of picture of control arm…
Bushing in question…
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.