Learn Farsi: باید سوپاپی را که مکانیکت شکست، تعمیر کنیم. - We must repair the valve that your mechanic broke.

The -i here is not the indefinite article meaning 'a/an'. It is actually the restrictive relative suffix. In Farsi, when a noun is followed by a restrictive relative clause (a clause starting with ke that defines exactly which item you mean), we attach this -i to the noun. So sopâp-i ke... literally translates to 'the specific valve that...'.

If the direct object is the entire phrase 'the valve that your mechanic broke', why does râ go right after sopâp-i instead of at the end of the whole clause?

This is exactly what this lesson focuses on! When a direct object is modified by a relative clause, the object marker râ is placed immediately after the antecedent (the noun being described) and its relative -i, wedging itself right before ke. While you might occasionally hear native speakers drag râ to the very end of the clause in sloppy everyday speech, sopâp-i râ ke... is the standard, correct syntax.

Is it normal to separate the modal bâyad and the main verb ta'mir konim by such a long phrase?

Yes, absolutely. Because Farsi is a Subject-Object-Verb language, the main verb (ta'mir konim) must wait until the very end of the clause. The modal bâyad typically sits at the beginning, meaning the entire complex object phrase—including its relative clause—gets nested right between them.

Does shekast mean 'broke something' or 'shattered on its own'? How do we know which one it is here?