• anujñātau — given leave
  Anu + √jñā — to permit. The grammatical hinge: Bhagavān moves only on the guru's anujñā. Same principle in Gauḍīya dīkṣā: nothing is done before guru-anujñā, after which everything is done.
• rathena — in (Their) chariot
  Ratha = chariot. The kṣatriya vehicle, taken up now that Their kṣatriya training is complete. Bhagavān returns to public history in His proper conveyance.
• anila - raṁhasā — of wind-like speed
  Anila = wind; raṁhas = speed. The chariot moves as fast as the wind. Hyperbolic but standard — kavi-samaya, poetic convention, here applied straightforwardly to Bhagavān's chariot.
• parjanya - ninadena — with cloud-like resounding
  Parjanya = rain-cloud; ninada = sound. The chariot rumbles like thunder. Same epithet later used for Kṛṣṇa's voice and for the conch Pāñcajanya — the home-return sounds like a coming storm of grace.
• sva - puram — (to) Their own city
  Sva-pura = own city — Mathurā. The chapter's geography closes: starting in Mathurā (v.1), going to Avantī (v.30), then Prabhāsa (v.38), then Saṁyamanī (v.42), and now returning to sva-pura. The circuit completes.

Receiving the guru's permission, Kṛṣṇa and Balarāma return home on a chariot that moves wind-fast and sounds cloud-like — a thunderhead of obedience rolling back to Mathurā. The journey home is announced by sound and speed because permission has been granted; without anujñā, even Bhagavān does not move.