Is this sentence ambiguous?

Slightly. "Give up", as you pointed out, would be very ambiguous. "Give *it* up" works well enough, and isn't likely to confuse most people. "Give *them* up" would be unambiguous.

To me it sounds weird. I think, because cigarettes is plural, it should be "give them up". If the sentence said "cutting down on smoking", then "give it up" is correct.

“Give it up” is a colloquialism, which means it might not be formally correct BUT is commonly used. I think give it up or give them up would be the most correct, but it looks like here they’re trying to get you to use directional words out of usual context, which is probably why it’s plain “up” in the assignment. In spoken English, “give it/them up” would be perfectly understood.

‘Give it up’ could technically work, but ‘give up’ (as in give up smoking) or ‘give them up’ work better. Cigarettes are plural, which makes ‘it’ not sound as natural. 

cutting back... give them up

It's not totally wrong but the exercise specifically asked for prepositions. Play the game by the rule!

The question should not be about ambiguity. It should be grammar check. Ambiguity is when the text is correct grammatically but the meaning is vagu

(NZ English.) I have no problem at all with "give up" as written in this context. I had to read the comments to see what the ambiguity might be. I get it now, but with one sentence saying they're cutting down, the next talking about giving up, and the next about stopping bad habits, no other meaning occurred to me. And grammatically, no pronoun is necessary.

Just putting "up" in that blank is hilariously wrong. *Don't say you "want to give up completely."* To give up completely is to abandon all hope and just stop trying. Whoops! You don't want to give up *everything*, you just want to give up smoking. So, you need the article so we know you're only giving up smoking. It is absolutely required here. If you said you were cutting down on smoking, then you'd say you want to give *it* up completely. But you said you were cutting down on cigarettes, plural, so you want to give *them* up completely.

Usually when someone says ‘give up completely’ it has a negative meaning it sounds odd when it’s left ambiguous since the subtext says that it’s supposed sound like you’re talking about giving up quitting (relapsing and smoking again), but the sentence doesn’t make logical sense unless it’s referring to the cigarettes.

the answer's wrong. someone saying it will be easier for them to "give up completely" almost reads like they want to kill themselves. "give it up" is also wrong, because what you're giving up is plural, cigarettes. "give them up" is correct. If the sentence said "I am cutting back on smoking at the moment, and I hope that this trip will make it easier for me to ..." then "give **it** up" would be correct. The reason is that "smoking" in this context would be kind of short for "my smoking habit", which is a singular noun ("my ... habit").

'Cigarettes' is plural and 'it' is singular. "give it up" therefore is a mismatch, it should be "give them up."

Yeah, while technically correct as just "give up" it sounds odd to me.

The problem with "it" is that you imply the habit, when it's never been stated before. Right answer would've been either "give them up" or just "give up".

"Give it up" is the correct answer. "Give up" could mean something different, and in any event, does not make sense based on the context. There isn't much nuance on this one, at least from American English.

If " to give up" is what's looked for here instead of "to give it/them up" then the sentence needs to be rewritten to sound natural at the very least. You could have "I hope this trip helps me give up smoking" etc but not the written sentence with only "up" in the blank (making "to give up completely") - is it understandable? Yes. Is it something a native speaker might say? Sure. Is it a good sentence? No.

It up is correct

Saying you will give "it" up feels weird here because "cigarettes" is plural. If it said "I am cutting down on smoking" then having "it" would make sense, but as it stands, I would say "give *them* up completely".

(UK English) It’s generally accepted that smoking is bad for your health and people should and do want to quit for that reason. “I’m trying to cut down smoking and hope to give up completely” is perfectly logical and standard English. To read “give up” as “give up cutting down” rather than “give up smoking” is technically possible but a bit perverse! The “it” isn’t technically wrong, but the use of “it” in “give it up” feels US-ian to me.