This is a decidedly un-Russian pronunciation. For one thing, those speakers are using their short British LOT vowel ɔ in the first syllable. But stressed Russian o resembles BrE THOUGHT oː far more. Here is Сочи uttered by several Russians:
In my experience, Russian speakers find it harder to approximate BrE LOT than THOUGHT. I sometimes mis-hear them as a result: for instance, I once thought a Russian was saying talking (which natives pronounce with the THOUGHT vowel) when in fact they intended topic (which natives pronounce with LOT). I haven’t noticed mis-hearings the other way round.
Of course spelling is a major reason for the English pronunciation. Written o doesn’t correspond to BrE THOUGHT before any consonant letter except r. THOUGHT would require spellings like Sawchi, Sauchi or (BrE being non-rhotic) Sorchi.
(The GOAT vowel might be another option for Sochi. My friend and colleague Prof. Masaki Taniguchi works at Kochi University in Japan, and my British colleagues and I pronounce Kochi with GOAT, like coach-y. Americans have GOAT in Sochi, and often use GOAT for orthographic o in foreign names where Brits prefer LOT, e.g. Prokofiev.)
Another difference between the English and Russian pronunciations of Sochi is the final vowel. Russian final unstressed i is pronounced ɪ, resembling the English KIT vowel. Here again are the Russian pronunciations of Sochi:
The Russian final vowel is nicely demonstrated by Russian-born American actor Anton Yelchin, who lays on a thick Russian accent as Chekov in the recent Star Trek films; here he says “Yes Sir, happɪ to”:
Classic Received Pronunciation used ɪ for the final vowel in happy, coffee, taxi, money etc., but contemporary SSB uses the FLEECE vowel. (In fact SSB disallows all short vowels in final position except schwa ə.) Here again are the British pronunciations:
There’s an easy way to help Russians who want to reduce their accent in English, namely by encouraging them to use a vowel more like their final unstressed –ij ий, as in Rimsky-Korsakov Римский-Корсаков:
Unfortunately the familiar transcription used in most dictionaries uses the symbol i in happy, coffee, taxi, money etc., more or less guaranteeing that Russians will mispronounce it, as they naturally interpret it as their final и. The default transcription used in the CUBE dictionary, e.g. happy hápɪj, generally gets better results.
To sum up. If you’re a BrE speaker who wants to pronounce Sochi in the most Russian manner that BrE vowels allow, you should put the THOUGHT vowel in the first syllable and the KIT vowel in the second. If you’re a Russian speaker who wants to pronounce Sochi like a BBC-type speaker, you need to make the first vowel much shorter and more open than the Russian vowel, and pronounce the final vowel more like -ий than -и.