折り入って

earnestly, sincerely

→ 折り入る
折り入っ Conjunctive te-form (linking form) + Te-form (conjunctive)
verb · imperative Rare #34,506
earnestly, sincerely Often used when making a serious or personal request, especially in 折り入ってお願いする (“to earnestly ask a favor”).
JA: 折り入ってお願いしたいことがあります。
EN: I have a personal favor to earnestly ask of you.

折り入ってお願いしたいことがあるのですが

I have a personal favor to ask you.

折り入って頼みたいことがある

There's something I'd like to ask you as a personal favor.

折り入ってお話がこんなやり方は無茶です

I have something important to discuss with you. This way of doing things is reckless.

— 8 1/2 (1963)

あなたに折り入ってお話したいことがあるのですが

There’s something important I’d like to discuss with you.

折り入って話がある

I have something important to discuss with you.

— レベッカ (1940)