I got an error:

from PyQt5.QtWebEngineWidgets import QWebEngineView
ImportError: QtWebEngineWidgets must be imported before a QCoreApplication instance is created

How to send data from Python to JS & back from JS to Python ...???

You can communicate between Python and JavaScript code in a QWebEngineView in PyQt5 using the WebChannel API.

To send data from Python to JavaScript:

1. Create a QWebChannel object:

channel = QWebChannel()

2. Add a Python object to expose to JavaScript:

class PyObj:
    def sendData(self, data):
        print(f"Received from JS: {data}")
        
obj = PyObj()
channel.registerObject("pyObj", obj)

3. Set the channel on the QWebEngineView:

view.page().setWebChannel(channel)

4. In JavaScript, access the Python object:

var pyObj = new QWebChannel(qt.webChannelTransport);
pyObj.objects.pyObj.sendData("Data from JS!");

5. The sendData() method in Python will be called.

To send data from Python to JavaScript, call a JavaScript function:

view.page().runJavaScript("jsFunction(data);")

And define the jsFunction() in JavaScript:

function jsFunction(data) {
  console.log(data);
}

Hope this helps!

Thank you.
updated code for PySide6

  <script language="JavaScript">
      
        new QWebChannel(qt.webChannelTransport, function (channel) {
          window.handler = channel.objects.handler;
          handler.test(function(retVal) {
            // console.error as console.log message don't show up in the python console
            console.error(JSON.stringify(retVal));
          })

          handler.send_to_server('hello')
        });
  </script>

import json
from PySide6.QtCore import QObject, Slot, QUrl, QJsonValue

class CallHandler(QObject):

  @Slot(result=str)
  def test(self):
      print('call received')
      return json.dumps({"abc": "def", "ab": 22})

  # take an argument from javascript - JS:  handler.send_to_server('hello!')
  @Slot(QJsonValue)
  def send_to_server(self, *args):
      print('i got')
      print(args)
      for arg in args:
          print(arg.toString())

Thanks, this helps a lot. I am using it to communicate with an embedded webapp inside python (inside the QWebEngineView). This was necessary because the webapp was sandboxed by the older engine inside QWebEngineView. So far I was able to restore Save/Load and retaining webapp internal settings after connecting the webapp and the python program.

https://github.com/Krambrek/inject

here a small enhancement with added demonstration of injecting JS into (remote) page and how to run JS on Python side.