Learn Polish: Proszę, oddaj tę kartę ojcu. - Please return this card to my father.

You are exactly right that most masculine nouns take -owi in the dative case. However, there are a few very common exceptions that take -u instead, and ojciec (father) is one of them. Also, notice how the e drops out of the root word, changing ojciec to ojcu. Other common masculine exceptions that take -u include pan (man/sir), chłopiec (boy), and pies (dog).

Why is the phrase tę kartę instead of tą kartę?

Karta is the direct object you are returning, so it must be in the accusative case. For feminine singular nouns, the accusative word for "this" is tę. Learners often mix this up with tą, which is the instrumental case form (used for tools or after prepositions like "with").

The English translation says "to my father", but I don't see the Polish word for "my" (mojemu) here. Why?

In Polish, you usually leave out possessive pronouns like "my" or "your" when the context makes it obvious. If you tell someone to give a card to "father", it is naturally assumed you mean your own father. Adding mojemu wouldn't be grammatically wrong, but omitting it sounds much more natural to Polish ears.

What form of the verb is oddaj?