There are a few different ways to approach it.

You can mock only foo using jest.spyOn and something like mockImplementation:

import { MyClass } from './MyClass';
import * as FooFactory from './somewhere/FooFactory';

describe('test MyClass', () => {
  test('construct', () => {
    const mock = jest.spyOn(FooFactory, 'foo');  // spy on foo
    mock.mockImplementation((arg: string) => 'TEST');  // replace implementation

    const c = new MyClass('test');
    expect(c).toBeDefined();
    expect(c.getState()).toEqual('TEST');  // SUCCESS

    mock.mockRestore();  // restore original implementation
  });
});

Similarly, you can auto-mock FooFactory with jest.mock, then provide an implementation for foo:

import { MyClass } from './MyClass';
import * as FooFactory from './somewhere/FooFactory';

jest.mock('./somewhere/FooFactory');  // auto-mock FooFactory

describe('test MyClass', () => {
  test('construct', () => {
    const mockFooFactory = FooFactory as jest.Mocked<typeof FooFactory>;  // get correct type for mocked FooFactory
    mockFooFactory.foo.mockImplementation(() => 'TEST');  // provide implementation for foo

    const c = new MyClass('test');
    expect(c).toBeDefined();
    expect(c.getState()).toEqual('TEST');  // SUCCESS
  });
});

You can also mock FooFactory using a module factory passed to jest.mock:

import { MyClass } from './MyClass';

jest.mock('./somewhere/FooFactory', () => ({
  foo: () => 'TEST'
}));

describe('test MyClass', () => {
  test('construct', () => {
    const c = new MyClass('test');
    expect(c).toBeDefined();
    expect(c.getState()).toEqual('TEST');  // SUCCESS
  });
});

And finally, if you plan to use the same mock across multiple test files you can mock the user module by creating a mock at ./somewhere/__mocks__/FooFactory.ts:

export function foo(arg: string) {
  return 'TEST';
}

...then call jest.mock('./somewhere/FooFactory'); to use the mock in the test:

import { MyClass } from './MyClass';

jest.mock('./somewhere/FooFactory');  // use the mock

describe('test MyClass', () => {
  test('construct', () => {
    const c = new MyClass('test');
    expect(c).toBeDefined();
    expect(c.getState()).toEqual('TEST');  // SUCCESS
  });
});