The city is Харків, or as the Russians spell it Харьков. If you're going to spell it Kharkiv, you must pronounce it 'Harkyoo [ˈxɑrkiw] Alex, not Har-kuv. Or possibly 'Harkeef [ˈxɑrkiːf] if you're using an eastern pronunciation. It's Ukrainian, not Russian.
For that matter, if you're using the Russian spelling of Kharkov, it's still not Kar-kuv; it's 'Harkoff [ˈxarʲkəf].
That H is really a voiceless velar fricative that we don't have as an initial consonant in English (think the CH in loch or Bach). But H is closer than K. And -uv is just wrong.
Stop saying you speak Russian, Alex.