I think I would just wait for the state to apply the amount to the return. OR call the state practitioner hotline and have them check to see if the amount has been applied. 

It's for last tax return (2020) and the change was only at the state level.

I think George's answer still holds.

Yes, amount applied and it brought down the taxes due around $500. Now, how can I just amend the state return?

Please go back and read George's post.

I don't know what you mean, you keep referring me to George respond. Client paid some amount with extension. then filed his return and gave me a wrong payment with extension for State, now that we figured it out we want to file state amendment because it brings his amount due around $500. I hope I was clear this time and I hope I geta better answer this time.

It doesn't get any better than what you got.  I give up.  Good luck with that amended return.

If you feel an urgent need to file an amended return for state only, go ahead. In Screen 1, remove all check marks except  for the state. In Screen 59, use the Ctrl + T to select the state to be amended and DO NOT CHECK Federal. Other than those two items, the article on amending should give you what you want. 

https://proconnect.intuit.com/community/individual/help/generating-form-1040x-amended-u-s-individual...

I still feel you should be able to call the state and get it adjusted. No amended return should be needed. The state is probably already doing the necessary steps to correct the issue. 

"The state is probably already doing the necessary steps to correct the issue."

The state has probably already fixed it.