Translation:Is there time?
In English, "Do you have time?" usually means an amount or period of time. (It is similar to asking, "Are you available?" or "Are you busy?")
"Do you have a time?" usually refers to a point in time. (It can be used to ask if the person knows/has decided on when something will happen. For example: Alice: "I'm going to a party on Friday! Do you want to come?" Bob: "I am busy until 3. Do you have a time?" Here Bob is asking if Alice knows what time the party is.)
The tips and notes give the example that 한 시 is "one o'clock" (a point in time) while 한 시건 means "one hour" (an amount of time), so I am assuming this sentence refers to an amount of time because it has 시간 as the subject. Therefore I believe that "Do you have time?" is correct and "Do you have a time?" is incorrect.
For more advanced students of English wanting to understand why the "a" makes such a difference in meaning: the word "time" has multiple meanings and it functions differently depending on the meaning. Specifically:
- Point in time, such as 1:00 PM. ("What time do you come home?") It is a countable noun for this meaning.
- A period of time with some sense of when it begins and ends. ("I was busy the whole time when my friends were at the party.") This I believe is countable and it is sort of an extension of the first meaning.
- Amount of time, such as an hour, without any implied start time. ("How much free time do you have this weekend?" "I could have finished the lesson in the amount of time I spent writing this comment.") This is an uncountable noun.
- An occurrence of something (often something that repeats). "I told you three times!" This is a countable noun.
Now recall the properties of uncountable nouns. Only countable nouns can have the indefinite article (a/an) and often require an article (the/a/an), but uncountable nouns cannot have the indefinite article (a/an) without units (and even then, only because units are countable nouns). Therefore, "Do you have a time?" cannot refer to a quantity of time because time in that sense is uncountable. It has to mean a point in time (e.g. 3:00 PM) or a specific period from one point in time to another (e.g. from 3:00 PM to 3:30 PM). (It could mean an example of an occurrence but that would be unlikely.) If you want to use "a" for an amount of time, you need units. "A minute" (or "a minute of time") is an amount of time and so is "three minutes" but not "a time" or "three times".
Other differences in usage also come from being countable or uncountable. For example, you use the words "much" and "less" with uncountable nouns, but you use "many and fewer" with countable nouns. Uncountable nouns normally are singular while countable nouns can be plural. So when talking about quantities of time, you can say "how much time" but with any of the other meanings (especially occurrence of something) you could say "how many times". You can also say "I spent less time learning than playing" (amount of time, uncountable), but "I told the truth fewer times than I lied" (occurrence of something, countable) or "I have fewer times on my calendar than you" (points in time, countable).