taru-adjective, adverb taking the 'to' particle
Describes a manner of speaking or explaining that is heartfelt, thorough, and patient. Often used with verbs like 諭す, 説く, or 語る. The word is a たる adjective/adverb taking と, so it typically appears as 懇々と.
The teacher earnestly admonished the student.
He spoke sincerely about his mistakes.