but rather though

04:20 Jul 24, 2020 

It's more a question of its suitability for inclusion in this sentence, in particular, where it comes-off as a bit clumsy. IMO. I would rather craft a source text that doesn't force high-register words into sentences where they really don't demonstrate a logical flow of ideas. In this case, "...no cost to you, but rather to ..." would require losing the parallel structure so commonly seen in good writing that uses these particular words. That is, there is no 'cost' referred-to in the second part of the sentence, as 'funding' is not a cost, as such.

I refer to this type of slip, if you will, as a stylistic 'infelicity' in my PhD thesis, in the section entitled, 'The translator as de-facto copy editor' (as part of my discussion of the potential roles of scholarly translators).

This is my take, at least. Cheers from Vietnam!

03:54 Jul 24, 2020 

It is eloquent in English because the emphasis is placed on "provision of X at no cost to you" but I agree with you, what is tricky is formulating the translation. The element of "sino [más bien]" is a good equivalent of "but rather", or perhaps "sino mediante".

https://www.grammarly.com/blog/comma-before-but/

03:49 Jul 24, 2020 

...at no cost to you, but rather [at a cost] to X ... ? It's a question of logical flow. It's not appropriate or very eloquently used in this sentence.

22:39 Jul 23, 2020 

Correction: This will be provided at no cost to you, but rather through funding provided by the NYS Office for Children & Family Services.

There is nothing wrong with this syntax in English. "But rather" goes with "provided." Instead of "you" paying, the NYS office pays. The only error is the omission of the "r" in "through."

03:43 Jul 21, 2020 

"This will be provided at no cost, through ..." y ya ;))