I think if you want to be considered for promotion, and your manager has said you're not ready because you need to learn some humility, then going and asking about promotion will illustrate their point very well!
If you're asking about promotion, then it says you think you're ready. They've told you they don't think you are.
I'd go to the manager, and not mention promotion at all. Say you've been thinking about what they said, and you're sorry if you have not come across well to clients. Ask if they can give you specific examples so you can work how you can do it better.
Perhaps ask them to observe an interaction and ask them to comment afterwards.
And (this is quite important) if they give comments, then don't justify or argue back.
Ask them how it would be better and take those comments and use them to make your own.
Keep asking for feedback, and keep adjusting depending on that feedback.
That will show humility, in that you are listening and trying to improve.
You want to avoid implying that you think you are the best/better. I know someone that will start a conversation by telling you that they are the best at whatever they're talking about even when they're clearly not. They're quite capable of standing there without a blush saying "and I know that I can do this better than anyone else here, because I'm superb at it" having read the first chapter of a book about it, and standing next to the person that invented the procedure and wrote the book!
The problem is when they do that in front of clients, often the client will accept their own valuation of themselves and expect a far higher output/result than they are capable of delivering. So the client is disappointed and reviews it poorly. Or someone else has to step in a clear up the mess. Whereas if they had been honest about it in the first place, expectations are not raised beyond what is possible to deliver.
That's why it can be important not to over sell yourself.