A humble expression used to preface a statement, objection, or request; literally 'with respect' or 'if I may say so' — used in formal or polite contexts.
恐れながら一つ申し上げますが、資料に誤りがあるようです。
If I may say so, I'd like to point out that there seems to be an error in the document.
恐れながら、別の案を提案させていただきます。
With all due respect, allow me to propose an alternative plan.
恐れながら先日の件についてはまだ確認が取れておりません。
I regret to say that I have not yet been able to confirm the matter from the other day.