Here's a few things that can cause the car to pull when accelerating -like you mentioned. I cut and paste the information and video. Hope this helps. Video was recent: March 2018.
In this video I discuss & show a car which pulls right when accelerating and left when decelerating so when pressing the throttle there is a large shift to the right in the steering direction, after letting off it pulls heavily back to the left as the load shifts.
There are a good few items that could cause this problem, to diagnose your issue correctly you will need to lift the vehicle and check each wheel & under your vehicle, check the control arms, the wishbones, ball joints, inner tie rods rack ends, track rod end, wheel bearings, steering rack & any other attaching component depending on your vehicles design.
This will allow you to find the play and fix the problem.
In cases like this it is important to determine the type of vehicle pulling that is involved, whether it is an alignment pull, a brake problem pull, a tire / tyre pressure pull, or like in this car a component has free play causing the pull.