Car gives a kick when transmission shifts hard?

Does the car have good engine power? If not, you could have a clogged catalytic converter which will cause the transmission to shift harder due to the lack of engine vacuum. If not, we need to run the codes to see what comes up.

Here is a guide to help you with the code gathering:

https://www.2carpros.com/articles/checking-a-service-engine-soon-or-check-engine-light-on-or-flashing

This guide can help tell if the cat is okay:

https://www.2carpros.com/articles/how-to-test-a-catalytic-converter

Please go over these guides and report back. If no codes come up, I would change out the torque converter turbine speed sensor. This can cause a problem without a trouble code. Check out the images (below). Please let us know what happens.

From the description it sounds like it could be missing a gear as it shifts. So, instead of going from first into second it's going from first into third. That would give you harsh engagement between gears. You might try changing the fluid in it but with 200,000 on it you are more likely looking at a transmission rebuild if the rest of the car warrants that.

This would give you an idea of the transmission service:
https://www.2carpros.com/articles/how-to-service-an-automatic-transmission