For people like me who live abroad, we go back home at\on this festival, but unfortunately, I wasn't able to do this at\on the last festival.
What if the festival was three days long? And what if it was one-day festival?
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Neither option is natural.
We can "go to a festival." If we want to specify the location of the festival, then we go to a location for a festival. Therefore, for is the correct preposition in both places in your sample sentence.
Your sample sentence also has some other more subtle problems. The construction "for (subject), (statement)" is helpful when the subject of (statement) is something other than (subject), and you want to express that the (statement) applies to that (subject). For example:
For people with disabilities, air travel can be difficult.
In your case, "people like me who live abroad" is the subject of the statement you're making. So the construction using "for" is not necessary at all, and you can say:
People like me, who live abroad, go back home for this festival.
You can also replace the last clause with something like:
... unfortunately, I wasn't able to do this last time (or last year, etc.).
we go back home at this festival ... I wasn't able to do this at the last festival.
This sounds like you are saying that when you go to the festival, you then go home. It doesn't make sense unless you are an alien and your spaceship is going to be at the festival to take you home.
we go back home on this festival ... I wasn't able to do this on the last festival.
This sounds like you are saying that when the festival happens, you go home. It's possible you could be going home when the festival is happening, but don't intend to go to the festival.
One of the meanings of X for Y is "X exists/is done for Y purpose/reason". So you should use this preposition if you are leaving home when the festival occurs since the festival is near your home and you intend to go to the festival.
we go back home for this festival ...
But since you didn't attend the last festival, the for connection isn't strictly needed. So on is OK here, but you could still use for if you wanted. You might still use for if you were contributing to the festival, e.g. if you worked there or were a performer.
I wasn't able to do this on/for the last festival.
It seems that "for" should be used after "go back home". For example,
They should go back home for the summer.
I'd Like To Go Back Home For Christmas.(it is the title of a book)
Consider these seven practical signs to determine if you can come back home for good.
I found these sentence from the internet, therefore it is highly likely that "For" is needed after".
Used to show an exact or a particular time:
-Are you free at lunchtime?
-The bells ring at regular intervals through the day.
-At no time/point did the company do anything illegal.
-I'm busy at the moment (= now) - can you call back later?
Used to show when something happens:
-Hair salons don't usually open on Sundays.
-What are you doing on Friday?
-My birthday's on 30 May.
-She was dead on arrival (= dead when she arrived) at the hospital.
So the result is:
For people like me who live abroad, we go back home on this festival, but unfortunately, I wasn't able to do this at the last festival.
in the first phrase we use "on" cause we are pointing out an event but in the second phrase we use "at" cause we are talking about a past event, a particular timeline.