Please; kindly; I beg of you — a humble, polite expression used when making a request (often in formal or business contexts).
本日は急なお願いで恐縮ですが、何卒よろしくお願い申し上げます。
I apologize for the sudden request today, but I humbly ask for your cooperation.
このプロジェクトへのご協力を何卒お願いいたします。
We kindly ask for your cooperation with this project.
何卒ご容赦ください。
Please forgive me; I beg your pardon.
By all means; without fail — used to express strong wish or insistence (somewhat formal).
お時間がありましたら、何卒お越しください。
If you have time, please do come (by all means).
何卒お見逃しなく！
Don't miss it — by all means!
この機会を何卒ご利用ください。
Please be sure to take advantage of this opportunity.