お～願（ねが）う is a 謙譲語（けんじょうご） ‘humble language’ expression in Japanese used for making requests in the same way as お～ください ‘please do (A)’. As with many other polite structures, the use of お will primarily appear with words of Japanese origin, while ご will be used for the majority Chinese-origin kanji compounds.
願（ねが）う itself is a う-Verb meaning ‘to petition’, and the literal translation is something like ‘I petition that you do (A) for me’, which sounds incredibly stiff in English, but gets the point across that someone is doing a favor for you.

• こちらからのメールが届（とど）いているか、お確（たし）かめ願（ねが）います。
  Could you please make sure that you have received our email.
• 鈴木（すずき）社長（しゃちょう）はもうすぐ到着（とうちゃく）するので、もう少々（しょうしょう）お待（ま）ち願（ねが）います。
  President Suzuki will be arriving shortly, so please wait a moment.
• この契約書（けいやくしょ）にご記入（きにゅう）された情報（じょうほう）に間違（まちが）いがないか、ご確認（かくにん）願（ねが）います。
  Please make sure that the information you have entered in this contract is correct.
• 歩道（ほどう）での禁煙（きんえん）にご協力（きょうりょく）願（ねが）います。
  We ask for your cooperation in not smoking on the sidewalks.

Naturally, due to being a polite sentence pattern, お～願（ねが）う will almost always appear in it’s ます-form, お～願（ねが）います.
Interestingly, words of western origin, such as katakana loanwords, will often omit both お and ご, appearing simply with 願（ねが）う attached to the end.

• この契約書（けいやくしょ）の下（した）の方（ほう）と、この書類（しょるい）の下（した）の方（ほう）にサイン願（ねが）います。
  Please sign at the bottom of this contract and at the bottom of this document.