Learn Hardcore Latin: Servus cito lucernam accendit, ut domina epistulam legere possit. - The servant quickly lights the lamp, so that the mistress can read the letter.

present: he lights
perfect: he lit / has lit The form looks the same.

The subject is usually shown by the nominative case. Here servus is nominative singular (the slave) and matches accendit (he lights / lit).
In the ut-clause, domina is nominative singular (the mistress) and is the subject of possit.

Why are lucernam and epistulam in the accusative case?

They are direct objects of their verbs:

lucernam (accusative) is what the slave lights: lucernam accendit.
epistulam (accusative) is what the mistress reads: epistulam legere.

What exactly does cito do in the sentence?

cito is an adverb meaning quickly. It modifies the verb accendit: the slave lights the lamp quickly.
Latin can place adverbs in several spots; here it’s placed early for emphasis and flow: Servus cito ... accendit.

Why does Latin use ut here, and what kind of clause is it?

ut introduces a purpose clause: = .A typical clue is without special “result markers” (like , , , etc.), so the natural reading is purpose.