The word “that” makes lots of people k-k-k-krazy.
“That” introduces a dependent clause, but only when it sounds right, which means, scarily, it’s a matter of how it sounds to you. There are no hard-and-fast rules, no absolutes. People hate that sort of thing. But it’s true. You’re more or less on your own when it comes to “that.”
“You will search books on English usage in vain for any uniform, much less helpful, guidance on the subject,” wrote Theodore Bernstein, former assistant managing editor of The New York Times, in his book Watch Your Language. Or, as Dr. Seuss put it, “You have brains in your head / You have feet in your shoes / You can steer yourself in any direction you choose / You're on your own / And you know what you know / You are the guy who'll decide where to go.”
That said, here are some things to consider:
AP Style says “When in doubt, include ‘that.’ Omission can hurt. Inclusion never does.” Sounds like good advice.
Note: To refresh your memory, dependent clauses are groups of words that contain a subject and verb but do not express a complete thought.