Expected a spy, but got Function

You need to get into the actual method, which in this case is on the prototype.

describe('When onGivePoints is fired', function () {
    beforeEach(function () {
        spyOn(UsersBoardCollection.prototype, 'restartPolling').andCallThrough();
        app.vent.trigger('onGivePoints');
    });
    it('the board collection should be fetched', function () {
        expect(UsersBoardCollection.prototype.restartPolling).toHaveBeenCalled();
    });
});

Spying on the prototype is a nice trick you can use when you can’t get to the actual instance you want to spy on.