Ah, OK, I see! Sorry, didn't get it at first.
Do you have multiple choice there or have to fill the gap by yourself?
As in, does it not accept an answer that it lists as a choice, or a "free" answer from you?
It wouldn't help much to give you an answer now because I actually don't know how these things work. It seems to me, gap-filling exercises are generated on the "backstage" in the Duolingo system, and are not written out by admins/mods. But I can ask around :)
Yes and no. In my bilingual household, the situation I would use this in, I would say "he's looking for his jacket" in english. When you say "він шукає куртку", it can mean either "the" or "his". If there is no subject previously established for "the" to refer to, it defaults to "his". For example, if everyone in the house is looking for a particular jacket and you ask what Dennis is doing, "він шукає куртку" means "he's looking for the jacket". But if we're all getting ready to leave the house and you ask what Dennis is doing, "він шукає куртку" means he's looking for his jacket. Possibly informal.
OK I got a bit confused here.
"He's looking for the jacket" can imply that the jacket is his, given a context. In the same fashion, "Він шукає куртку" can imply that too. However, the sentence does not explicitly contain the word "his/свою".
So, "He's looking for his jacket" is not really a translation of "Він шукає куртку" because its meaning is more specific than that of the Ukrainian sentence which is more general/vague/ambiguous and vice versa. As another example, "He's looking for the jacket" can also mean "He's looking for her jacket" in a specific situation. But that doesn't mean we can translate "He's looking for the jacket" as "Він шукає її куртку". True, some contexts are more common than others (in this case, "his jacket" is a more common situation). But the sentence is context-independent, and by adding extra context it's not a translation anymore, it's interpretation.
At least that's my perception of it :)