Learn Turkish: Sekreter röntgen için başka bir kliniğe gitmemizi söyledi. - The secretary told us to go to another clinic for the x-ray.

This is the core of an indirect command! It is formed by the verb root git- (go), the short verbal noun suffix -me (making it a noun-like action, similar to "going"), the possessive -miz (our), and the accusative case marker -i. Literally, it translates to "our going."

Why do we need the accusative case -i at the end of gitmemizi?

In Turkish, verbs like söylemek (to tell/say) take a direct object for the thing being said. The secretary told what? She told "our going" (gitmemiz). Because this entire action is the specific direct object of the verb söyledi, it requires the accusative suffix -i.

Why doesn't the sentence include bize (to us) if she told us?

You could say bize söyledi (told us), but it is usually left out. The possessive suffix -miz (our) in gitmemizi already makes it completely clear that we are the ones who are supposed to go, so adding bize is redundant unless you really want to emphasize it.

What happened to the spelling of the word klinik in this sentence?