tldr; use a JSON library - I like Json.NET - and don't worry about it.
The JSON shown is valid JSON where \uABCD in a JSON string represents a UTF-16 encoded character1. The internal JSON character escaping format is useful to avoid having to deal with Unicode stream encoding issues - it allows JSON to be represented entirely in ASCII/7-bit-clean characters (which is a subset of UTF-8).
Using a conforming JSON library to parse the JSON with such escapes would restore the JSON into an appropriate object-graph, of which some values will be properly-decoded String values. The library is responsible for understanding JSON and converting/reading it as appropriate - this includes correctly handling any such \u escape sequences.
The stream itself (that of the JSON text) should use the encoding that the server says, is indicated by a BOM, or has been pre-negotiated: but really, just UTF-8 here. This is how the JSON text is encoded, but has no bearing on the escape sequences found in JSON strings.
1 Per RFC 4627, The application/json Media Type for JavaScript Object Notation (JSON):
Any character may be escaped. If the character is in the Basic
Multilingual Plane (U+0000 through U+FFFF), then it may be
represented as a six-character sequence: a reverse solidus, followed
by the lowercase letter u, followed by four hexadecimal digits that
encode the character's code point. The hexadecimal letters A though
F can be upper or lowercase. So, for example, a string containing
only a single reverse solidus character may be represented as
"\u005C".
Alternatively, there are two-character sequence escape
representations of some popular characters. So, for example, a
string containing only a single reverse solidus character may be
represented more compactly as "\\".
To escape an extended character that is not in the Basic Multilingual
Plane, the character is represented as a twelve-character sequence,
encoding the UTF-16 surrogate pair. So, for example, a string
containing only the G clef character (U+1D11E) may be represented as
"\uD834\uDD1E"
For the doubters, here is a LINQPad example. This uses JSON.Net and imports the Newtonsoft.Json.Linq namespace.
var json = @"{""name"":""K\u0131r\u0131nt\u0131 Reklam""}";
json.Dump(); // -> {"name":"K\u0131r\u0131nt\u0131 Reklam"}
var name = JObject.Parse(json)["name"].ToString();
(name == "Kırıntı Reklam").Dump(); // -> true