Pray let your servant return: in this case your servant is used by Barzillai to refer to himself. But translators must be careful, since the same words are used by the same person in the next sentence to refer to Chimham. The request to return home is made in very polite and respectful terms. And the same degree of politeness should be evident in the receptor language if at all possible.
In my own city: that is, “at home,” “in my hometown,” or possibly “at the place where I came from.”
Here is your servant: literally “behold your servant” (compare King James Version). In this case your servant refers to Chimham and not to Barzillai. This is a way of presenting Chimham to David. Translators should look for the most natural way for someone to present a servant or worker to a person in authority. In some cases one may say “I am giving my servant Chimham to you…” or “here is my servant Chimham. He now belongs to you.”
Scholars often take Chimham to be Barzillai’s son, but this is not clearly stated in the Bible. It would, however, be normal for an elderly person to offer his son to do something for a king when the older person is unable or unwilling to do what the king requests. Also, the “sons of Barzillai” are particularly recommended by David to his son, Solomon, at the end of his life (1 Kgs 2.7). Some manuscripts of the Septuagint say “my son Chimham,” and this is the basis for the rendering “Here is my son Kimham” (Revised English Bible, Bible en français courant). Other translations such as Good News Translation, Die Bibel im heutigen Deutsch, Parola Del Signore: La Bibbia in Lingua Corrente, Biblia Dios Habla Hoy, and Nueva Biblia Española also say “my son,” but it seems that this addition is based on an interpretation of the Hebrew text rather than on the Septuagint. Such an interpretation is probably correct and may serve as a model for other translations.
The verb go over means more than merely crossing the river. In this context what is involved is accompanying David all the way to Jerusalem and continuing to serve him after arriving there.
Whatever seems good to you: literally “whatever good in your eyes.” Some other renderings are “whatever pleases you” (New International Version) and “as you see fit” (New Jewish Publication Society’s Tanakh). Compare the similar expression in verse 18.
Quoted with permission from Omanson, Roger L. and Ellington, John E. A Handbook on the First and Second Books of Samuel, Volume 2. (UBS Helps for Translators). New York: UBS, 2001. For this and other handbooks for translators see here .

I found this very interesting, in that first he is called Kimham, then Chinham – but I see it as a master giving the best servant has to the king. (As we do not know for sure if the servant is Barzilii’s son. Saying, “Do as you wish with him,” meaning bless him/reward him, for he is as faithful as if he were my own son. In so doing he is blessing King David, as well as the servant offering him a better life going forward in the palace. One could preach from many aspects of this text.