Here is an example of how you might go about it.
The feature layer myPoints on the map has a stack of points at Durham, NC (along with various other points at other cities).
Using the selection tool, I selected the stack of points. You can see in the attributes that there are several selected features.
In a Notebook in ArcGIS Pro, I specify the layer on the map I want to work with and assign it to the variable "fc".
I make a list of the fields I'm interested in using, SHAPE@X and SHAPE@Y.
I run a Search Cursor on the selection to print the coordinates. I also grab the coordinate values and assign them to variables to use later (when the loop stops, the values are those of the last point).
In the next cell of I my notebook, I use an Update Cursor to move each of the points to make a diagonal line of points.
I take the starting values from the variables I grabbed for the coordinates in the previous cell.
In the cursor, I set row[0], the value of X coordinate, to the value of the X coordinate I grabbed before. Then I update that value by adding -0.0001 degrees (this data is in decimal degrees... if your data is in planar units you would add a planar distance, like 1 meter, or 10 feet).
I also do the same thing with the row[1] value, for the Y coordinate (adding a similar amount).
Then I update the row (the current feature in the cursor) to use the new values, and the loop happens again, for each of the selected features.
The result is a diagonal line of points starting at the original location.
This is a little dangerous, as it is altering your data. I would strongly recommend making a copy to test on. If you forget to select a specific stack of features and run the code, the code will run on all of the points in your feature class, and all 13K points (including good mine locations) will be rearranged into a nice line (and the code might fail when the values get too large).
It does show how you can manipulate point geometries with Python.