Count: 0 — There are zero Russian headwords that begin with the Latin letter H. Russian is written in Cyrillic, and standard transliteration systems such as BGN/PCGN map the Cyrillic letter Х to “Kh”, not “H”. Check authoritative sources (Russian National Corpus, major dictionaries, Wiktionary) and you will find entries that start with Cyrillic Х (for example хлеб → khleb, Хабаровск → Khabarovsk) but not with a Latin H. Note that Ukrainian uses a letter and sound often transliterated as H, so Ukrainian word lists will include many H‑initial entries while Russian lists will not.

Understand the reason: modern Russian lacks a native /h/ phoneme, so historical sound changes and borrowing practices produce /x/ or /ɡ/ instead, and transliteration conventions preserve that with “kh” or “g”. Expect close matches among Russian words that start with Cyrillic Х and among foreign names rendered into English through other languages. For practical results, consult a list of Russian words beginning with the Cyrillic Х, or look at Ukrainian H‑initial words if you need Latin‑H starts.