Jest spyOn function called

You were almost done without any changes besides how you spyOn.
When you use the spy, you have two options: spyOn the App.prototype, or component component.instance().

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const spy = jest.spyOn(Class.prototype, “method”)

The order of attaching the spy on the class prototype and rendering (shallow rendering) your instance is important.

const spy = jest.spyOn(App.prototype, "myClickFn");
const instance = shallow(<App />);

The App.prototype bit on the first line there are what you needed to make things work. A JavaScript class doesn’t have any of its methods until you instantiate it with new MyClass(), or you dip into the MyClass.prototype. For your particular question, you just needed to spy on the App.prototype method myClickFn.

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jest.spyOn(component.instance(), “method”)

const component = shallow(<App />);
const spy = jest.spyOn(component.instance(), "myClickFn");

This method requires a shallow/render/mount instance of a React.Component to be available. Essentially spyOn is just looking for something to hijack and shove into a jest.fn(). It could be:

A plain object:

const obj = {a: x => (true)};
const spy = jest.spyOn(obj, "a");

A class:

class Foo {
    bar() {}
}

const nope = jest.spyOn(Foo, "bar");
// THROWS ERROR. Foo has no "bar" method.
// Only an instance of Foo has "bar".
const fooSpy = jest.spyOn(Foo.prototype, "bar");
// Any call to "bar" will trigger this spy; prototype or instance

const fooInstance = new Foo();
const fooInstanceSpy = jest.spyOn(fooInstance, "bar");
// Any call fooInstance makes to "bar" will trigger this spy.

Or a React.Component instance:

const component = shallow(<App />);
/*
component.instance()
-> {myClickFn: f(), render: f(), ...etc}
*/
const spy = jest.spyOn(component.instance(), "myClickFn");

Or a React.Component.prototype:

/*
App.prototype
-> {myClickFn: f(), render: f(), ...etc}
*/
const spy = jest.spyOn(App.prototype, "myClickFn");
// Any call to "myClickFn" from any instance of App will trigger this spy.

I’ve used and seen both methods. When I have a beforeEach() or beforeAll() block, I might go with the first approach. If I just need a quick spy, I’ll use the second. Just mind the order of attaching the spy.

---

EDIT:
If you want to check the side effects of your myClickFn you can just invoke it in a separate test.

const app = shallow(<App />);
app.instance().myClickFn()
/*
Now assert your function does what it is supposed to do...
eg.
expect(app.state("foo")).toEqual("bar");
*/

EDIT:
Here is an example of using a functional component. Keep in mind that any methods scoped within your functional component are not available for spying. You would be spying on function props passed into your functional component and testing the invocation of those. This example explores the use of jest.fn() as opposed to jest.spyOn, both of which share the mock function API. While it does not answer the original question, it still provides insight on other techniques that could suit cases indirectly related to the question.

function Component({ myClickFn, items }) {
   const handleClick = (id) => {
       return () => myClickFn(id);
   };
   return (<>
       {items.map(({id, name}) => (
           <div key={id} onClick={handleClick(id)}>{name}</div>
       ))}
   </>);
}

const props = { myClickFn: jest.fn(), items: [/*...{id, name}*/] };
const component = render(<Component {...props} />);
// Do stuff to fire a click event
expect(props.myClickFn).toHaveBeenCalledWith(/*whatever*/);

If a functional component is niladic (no props or arguments) then you can use Jest to spy on any effects you expect from the click method:

import { myAction } from 'src/myActions'
function MyComponent() {
    const dispatch = useDispatch()
    const handleClick = (e) => dispatch(myAction('foobar'))
    return <button onClick={handleClick}>do it</button>
}

// Testing:
const { myAction } = require('src/myActions') // Grab effect actions or whatever file handles the effects.
jest.mock('src/myActions') // Mock the import

// Do the click
expect(myAction).toHaveBeenCalledWith('foobar')