Could anyone tell me what  the differences between 'single' and 'only' are?

Dear Helpers,

I don’t know when I can omit the word “that.” I looked for the word “that” in the index of my grammar book.  I found it in different chapters.

Case ( A ) : Object of a verb:
The man (that) I saw told me to come back today.
This is the best hotel (that) I know.

Note:
I read “If it is the object of a verb, the “that” can be removed.”

Case ( B ) :The conditional:
I hope (that) I will succeed.
I hoped (that) I would succeed.
He thinks (that) they will give him a visa.
He thought (that) they would give him a visa.

Note:
“that” are in parentheses in the book.

Case ( C ) :Reported speech
He explained that he never ate meat.
He said he was waiting for Ann.
He said he had found a flat.
He said that Ann would be in Paris on Monday.
Peter said that they ought to widen the road.
Bill said he would be 21 the following day.

 
Case ( D) :Noun clauses introduced by that:
It occurred to me that he might be lying.
It appears that we have come on the wrong way.
It is a pity that he didn’t come earlier.
He was relieved that no one had been hurt.
I’m delighted that you can come.

I understand that in the cases A and B, the word “that” can be omitted. I don’t know about the cases C and D.  Can  I omit the "that?"

Thanks in advance....

Spoonfedbaby
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