"That" is a pronoun used for identification.
"What is that?" (accompanied by pointing)
"Where is that?" (referring to a location just mentioned)
"It's that time of day again." (referring to the current time and some generalization about it)
"That" should only be used when a more specific noun is lacking or not worth repeating. For that reason (did you really want me to repeat the previous sentence?) it is often used when inquiring as to what the noun should be. The above examples, for instance, are each expecting a noun or an explanation in return.
If you're familiar with programming concepts, consider "that" as a variable that can be assigned almost any value based on context and inferred meaning.
Alternatively, consider it as a box in which you can place a word, concept, idea, or anything else that can be referenced. Once the thing is in the box, you can simply refer to the box to make the communication more efficient. The only requirement for what can be placed in the box is that the receiver knows what was placed in the box. (In the previous sentence, "that" was used to signify that what followed was being assigned to the word "that". Try the sentence without it and you may notice the lack of clarity.)
Does that make sense? (again reducing the word count by using the "that" box to refer to the last thing present, in this case the previous sentence)
In your example sentence there are multiple redundancies. It could be simplified to, "Is it correct to use the word that in this manner?"
"My question", "I'm asking", and "is (followed by a question mark)" all communicate the same thing, and "is" has the fewest syllables making it more efficient. The phrase, "the word" is an attempt to clarify that the following word is the subject.