What is the grammar of "the hope is..." versus "the hope is that"?

Question

The sentence in question:

But with the US leading by example, the hope is that international efforts to tackle global warming will be revitalised.

Source

I'm a non-native speaker. I understand that natives use "hope is" like this:

With hard studying, hope is I will pass the test.

Please break this down grammatically for me to understand. In my understanding, the sentence is grammatically wrong because it should look like "A is B".
"A is B will C" is not an acceptable form.

I couldn't find any grammar rule pertaining to "hope is~".
Is there a "that" (relative pronoun) dropped somewhere?
Or does the word "hope" have some hidden grammar skillset/quality that most non-natives don't know about?

I think I found the grammar error first, then BBC had since updated the post. (Proof : https://hellotalk.com/m/uPyWJUS3OMuFDD==?id=L0pvNuN=)

Anyway, so even after BBC's update everybody here is saying that "that" can be omitted so I'm researching the conditions for omission of relative pronoun that. BTW, if you guys could just name the concise and precise law of the grammar that allows for this, things would be simpler. I had ran into another grammar question before, and to understand this properly I had to be insistent in my pursuit. (https://english.stackexchange.com/questions/513066/havent-we-vs-dont-we)

Just saying, contextually understood grammar technically isn't grammar. Grammar is rules and structures, not a literature piece.

That can be omitted when

1. it's pointing to an object (the girl (that) I love)
2. that after some emotional conviction words (I am glad that you came.)
3. after an attribution verb (Tom said that he didn't feel well.)
4. phrasal expressions (You may play provided that you do your homework first.)

Answer 1

You are correct that "A is B will C" is not an acceptable form. However, that is not the form of the BBC sentence. The form is actually "A is B":

• A is the nominal phrase "the hope" (which functions as a subject).
• B is the nominal clause "that international efforts to tackle global warming will be revitalised" (which functions as a predicate nominative).

In the sample sentence that you wrote, the nominal clause "[[that]] I will pass the test" functions as a predicate nominative. As you realized, "that" has been omitted.

I'd encourage you to research nominal clauses if you're not familiar with them. (Some people also say "content clause", "that-clause", etc.)

By the way, "hope" requires a determiner, so you should say "the hope is . . .", "our hope is . . .", etc.:

With hard studying, my hope is [[that]] I will pass the test."

Answer 2

"With hard studying, hope is I will pass the test" is a colloquial form that I would not say.

A fuller form of the same sentence is

The hope is, I will pass the test.

or even more fully

The hope is that I will pass the test.

This has the same structure as the relevant part of the sentence you are asking about.

(I would probably say "My hope is, I'll pass the test".)

Does that make it clear or is there something you still don't understand.