I found in a book the sentence "We would ask you to clear the balance immediately." Is this correct English or should it rather be: "We would like to ask you to clear the balance immediately"?
You're asking about this sentence:
We would ask you to clear the balance immediately.
It appears from comments to your question that you're wondering whether "would ask" works there, as opposed to "would like to ask".
You also mention the topic of politeness in the title of your question. Padding out the basic "Clear the balance immediately." with little words certainly fits in with diplomatic language:
- Use little words to soften your statements - 5 Tips for Polite and Diplomatic Language
The word would can be used in the following ways:
would modal verb 3 Expressing a desire or inclination. ‘I would love to work in America’ 4 Expressing a polite request. ‘would you pour the wine, please?’ - ODO
I think both definitions apply here. The intended request is:
- We ( ) ask you to clear the balance immediately.
Adding the word would softens this somewhat. It might look as if it is simply indicating that at some stage in the future, the above request would be made (just as your "would like to" variant does). However, in BrE especially, and perhaps in other dialects, the context might dictate that this is the request, just delivered politely. Here's another example in print:
All are welcome and admission is free. This is one of Dr ...'s rare public appearances in the UK. In order to secure your place at the lecture, we would ask you to register in advance by filling in the form below and returning it before 24 February. ... - a sample quoted in The Pragmatics of Translation, edited by Leo Hickey
The book goes on to identify problems with mixing genres when that text used as an 'academic announcement' rather than as 'show business publicity material, but the use of "we would ask you" passes without comment.
Here's another example:
To ensure everyone enjoys their day supporting the club we'd like to take this opportunity to remind fans that they are representing Middlesbrough Football Club at all times, and we would ask you to help us in ensuring the Riverside is a safe place for all. - Middlesbrough Football Club is issuing a polite reminder to fans that the use of pyrotechnics at the Riverside Stadium is not allowed
To sum up: the original works grammatically. It has the same sense as your alternate "would like to" wording.