Correct spelling, explanation: hoping is a present participle derived from the verb hope. There is the rule that says that you should add the suffix ing to the base hope and adjust the spelling, in this case, reduce the vowel e like in words: smoke – smoking, settle – settling, make – making. It means that e is a non-syllabic vowel sound and therefore it should be removed. It became silent in late Middle English or Early Modern English. The first known use of hope was before the 12th century but it has its roots in Old English as hopian. Thus, the form hoping is written correctly.

Definition of hoping:
verb, a present participle form from a verb meaning to expect something or want something to happen,
I am still hoping to see my friend soon.
She bought tickets for the play, hoping that they watched it in the theatre.

Phrases with hoping:
here’s hoping,
Here’s hoping that the book I have borrowed from you is interesting.
hope so – used to say that someone wants something to be true,
Will he be able to come to the meeting? I hope so.
hope not – used to say that someone wants something not to be true,
Will he have to miss the meeting? I hope not.

Incorrect spelling, explanation: hopeing is an incorrect form of present participle that is derived from the verb hope. Admittedly, the rule says that we should add the suffix ing to the base form but it is required to remove the silent vowel e. Therefore, hopeing is incorrect, only the form hoping exists in English.