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.
Could you please make sure that you have received our email.
President Suzuki will be arriving shortly, so please wait a moment.
Please make sure that the information you have entered in this contract is correct.
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
, appearing simply with 願（ねが）う
attached to the end.
Caution - As お～願（ねが）う is an even more polite way of saying お～ください, or simply even just てください, it will rarely be used outside of formal circumstances, such as voice-overs giving instructions etc. If used in daily speech, it may come across as sarcasm.