I dont know whether to use singular or plural case here. I couldnt find an answer to this quickly browsing the web and could argue for both being correct.Maybe they are but do they have slightly different meanings then?
It seems like you're reaching for the correct idiom to use with set and hope. As with most English idioms, there is no underlying rule and you simply have to memorize the pattern, which is "to set ones hopes on ..."
The candidate set his hopes on winning the debate
She set her hopes on getting the big promotion
They set their hopes on landing the contract before Christmas
This idiom means not only that you hope some outcome will happen, but that in some way you rely on it happening -- that there could be some consequence if it didn't happen.
Why "hopes" and not "hope"? Again, idiom is the way it is and while we could probably create a logical explanation, I doubt it would explain anything else. Anyway, your sentences are grammatically fine, but the meaning doesn't parse as well as if you use the idiom.
Of course you can just use the verb "to hope" in the usual way, to simply imply that you want something to happen:
The candidate hoped to win the debate
She hoped to get the big promotion
They hoped to land the contract before Christmas
Another related idiom is "to have high hopes" This means that you really want something to happen, so much that you almost expect it to happen, but not necessarily that you rely on it happening.
The candidate had high hopes that he would win the debate
She had high hopes for getting the big promotion
They had high hopes they would land the contract before Christmas
Which you choose depends on what, exactly, you want to say.