Learn Finnish: Agentti luki kirjeen ymmärtääkseen salaisen tehtävän. - The agent read the letter to understand the secret mission.

This is the structure you are currently learning. It is formed by taking the basic dictionary form of the verb ymmärtää (to understand), adding the translative ending -kse-, and finishing with a possessive suffix. Because the agent (he or she) is the one understanding, we use the third-person suffix -en. Altogether, ymmärtääkseen means "in order for him/her to understand."

Why are the objects kirjeen and salaisen tehtävän both in the genitive case, ending in -n?

They are both total objects. The agent read the whole letter (kirjeen is the object of luki), in order to understand the entire mission (salaisen tehtävän is the object of ymmärtääkseen). Because the sentence has a clear subject (agentti) performing the actions, these objects take the normal -n ending to show the actions are complete.

Could I use the word jotta (so that) instead of the -kseen structure in this sentence?

Yes, absolutely! The -kseen structure is a compact, slightly more written-language way to express purpose. You could rewrite the second half of the sentence using a jotta clause and the conditional mood: ...jotta hän ymmärtäisi salaisen tehtävän (...so that he would understand the secret mission). Both mean exactly the same thing.

What if the subject was "I" instead of "the agent"? How would ymmärtääkseen change?