"I do not like to travel with my uncle."
Translation:Я не люблю подорожувати з дядьком.
Ukrainian nouns and adjectives have several case forms. The adjective must have the same case as the noun it modifies. After «з» (z), we use Instrumental case, so you should use з моїм дядьком (I'm not sure how it's rendered in Duolingo's Latin script, maybe z moyim dyad'kom?).
Мій (miy) is a different case, it's nominative. It's used for the subject of the sentence: Мій дядько — вчений (Miy dyad'ko - vchenyy) 'My uncle is a scientist'.
Using different case for adjective and noun (z miy dyad'kom) is definitely wrong. Although you'd be understood, of course, so don't worry if you use wrong forms in speech. :)
Duolingo often allows several possible variants of the same sentence, so if it's not listed as the suggested translation, it doesn't mean it's not accepted.
In general, I think this sentence sounds better without 'my' in Ukrainian. In English, we just can't leave the word without a modifier (we need to add either 'my', or 'the', or 'a', and 'my' sounds most natural). Ukrainian, however, doesn't require any modifiers here, and if you don't specify it's your uncle, it's still understandable you're taking about your own uncle.
There are many more examples when Ukrainian doesn't require possessive pronouns when English does, e.g. я ми́ю ру́ки 'I wash my hands'. In English, it sounds unnatural without 'my'. In Ukrainian, on the other hand, it sounds unnatural if you add 'мої' (because it's pretty obvious you're washing your own hands).
1) In Ukrainian, if the possessive pronoun refers to the subject, "свій" is used. He ... with his... -> Він ... з своїм ...; I ... with my ... -> Я ... з своїм ...; She ... with her ... -> Вона ... з своїм and so on.
2) I don't know why, but here using Я з моїм is accepted. Some sentences are like this by default, and I cannot change them. They wrote them this way, and recorded the audio this way. I have no idea why. Maybe they thought it's going to be less confusing to foreigners. So, for the sake of consistency, I think it should be accepted everywhere else then. So I added it here.
Now the accepted versions are з моїм дядьком, з дядьком and зі своїм дядьком.