問題一:請問由that引導的名詞子句，做同位語時，可以省略that 嗎？有些人說可以，有些說不能，請舉幾個例子說明。問題二:為什麼前面有介系詞時，that不能用? 感謝您！?

問題一:可以的.如下例子們...

Trump refused to recognize (that) man-made global warming is true.
Stop telling me (that) you love him.

題二:
因為＜介系詞＋that＞時常有特殊的意義
例如in that = because

She stops coming to the club in that she got preggers.

問題一:
由that引導的名詞子句 (noun clause) 像一般名詞一樣，可以當主語、賓語或補語，當然亦可當同位語。除了當賓語和補語(形容語或主語)可以省略外，其餘(包括同位語)都不能省略。
***Note: "that" after nouns wouldn't normally be omitted *****

例:
1. object (of reporting verbs)
She said (that) the story was true.
We recognized (that) the task was not straightforward.
*** exception: "that" would not normally be omitted after verbs such as reply, shout, etc.
2. adjective complement
It’s possible (that) he has not received the letter.
3. subject complement
The problem is (that) you never think before you speak.
4 subject
That he is your brother-in-law should not affect your decision.
5. object complement
I declare the problem that you do not want to learn about grammar.
6. appositive
The problem, that you did not pick up the packages, delays the entire production schedule.

題二:為什麼前面有介系詞時，that不能用?
When "that" functions as a relative pronoun, it cannot follow a preposition. Instead, the preposition has to be placed at the end of the relative clause, or otherwise another pronouns "which" or "whom" should be used.

Although noun-clauses can act as the object of preposition, the noun clauses introduced by conjunction "that" cannot follow a preposition, based on the grammar rule that prepositions cannot be followed by conjunctions.. As that-clause normally refers to an abstract thing, such as a fact, an idea, a problem, etc., the sentence will be grammatical if we insert this noun before the that-clause.

example -
I knew about his problem.
His problem was that he ran out of money.
- I knew about that he ran out of money. (X)
- I knew about his problem that he ran out of money. (O)
- I knew that he ran out of money. (O)
- I knew about the fact that he ran out of money. (O)