Mocha motion tracking data offsets/clips the layer I want to apply it to

Something is scaled or moved in the layer that was tracked or in the layer to which you applied the tracking data. 

I suggest that you go through the Mocha Tuturals from Boris FX. 

My standard getting ready to track using Mocha:

1. Trim the footage layer to only the frames that need tracking (split it if you have to)
2. Apply color correction to improve contrast and edge detail; it will be turned off later
3. Pre-compose the layer, moving all attributes and trimming the pre-comp to the length of the layer
4. Apply Mocha to the nested Pre-comp you just created
5. Run the tracking, masking, and other data you need using Mocha
6. Make sure that you preview any surface, tracking, or mattes you are creating in Mocha
7. Save/Return to AE/apply the data you created
8. When completed, open the pre-comp you created and turn off the color correction you added to improve tracking accuracy

That should get you started. 

In the future, when you post screenshots, include the UI and show the modified properties of the layers you are having problems with. It is awfully hard to draw any meaningful conclusions about your workflow problems if all we see is the final image.

Maybe this tutorial I did will help you figure things out. I did forget to pre-compose my trimmed layer in the tutorial, but you should get the idea.

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Shape layers calculate their coordinates from the center, not the top left corner like otehr layers or for that matter mocha. The immediate solution therefore would be to use a standard solid, not a shape layer. Otherwise it could of course be fixed by recalculating the coordinates with an expression or massaging them in mocha.

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