Incorrect spelling, explanation: there is no such form as hopeing because when the gerund is created from the verb, ending at the non-syllabic vowel sound e, it should be removed, and then the suffix ing is added to the base. Hence, hopeing is written erroneously, and only hoping is without the mistake.

Correct spelling, explanation: the word hoping is a gerund, forming from the verb hope that already existed in Old English as hopian. The verb hope was first used in Middle English. And then the vowel sound e, occurring at the end of a syllable, became silent. Making the gerund, it is required to reduce this non-syllabic vowel e and next to add ing ending. As a result, hoping is spelled errorless and hopeing is a misspelling.

Definition of hoping:
verb, (gerund from the verb hope) to have the feeling that what is wanted and expected will happen,
Maria has bought a backless one-piece swimsuit, hoping that she will go to the seaside holiday.
He insisted, still hoping to get a positive answer.

Phrases with hoping:
hoping for – looking forward to something,
David is hoping for an interview on Friday.
hope that something sticks – want something to be successful, especially a business strategy,
The new food service they’re trying out is pretty good, so they hope it sticks.