adverb
earnestly; sincerely; with all one's heart
Used when making a heartfelt request or expressing a strong, sincere desire. Often paired with お願い, 頼む, or similar verbs.
折り入ってお願いしたいことがあります。
I have a favor to ask of you earnestly.
折り入ってご相談したい件がございます。
There is a matter I would like to discuss with you sincerely.
Derived from the verb 折り入る (to bend inward, to be deeply involved), with the て-form used adverbially. The exact historical development is uncertain, but the expression conveys a sense of deeply and sincerely engaging in a request.