Στο άλλο υποκατάστημα μού ζήτησαν άλλο ένα δικαιολογητικό και ένα αποδεικτικό διεύθυνσης.

Στο is the contraction of σε + το.

• σε = in / at / to
• το = the for a neuter singular noun

So στο άλλο υποκατάστημα means at the other branch.

After σε, Greek uses the accusative, which is why the noun phrase appears in that form.

The article makes it specific.

• στο άλλο υποκατάστημα = at the other branch
• σε άλλο υποκατάστημα = at another branch

So the sentence is talking about a specific different branch, not just any other branch.

Also, άλλο agrees with υποκατάστημα:

• υποκατάστημα is neuter singular
• so the adjective is also άλλο (neuter singular accusative/nominative form)

Μού means to me or, in natural English here, simply me.

It is the weak form of the first-person singular pronoun (μου) and is grammatically a genitive clitic pronoun. With verbs like ζητώ, Greek often uses this form for the person involved:

• Μού ζήτησαν... = They asked me for...

A fuller version would be something like ζήτησαν από εμένα, but Greek usually prefers the short clitic form.

Because Greek weak object pronouns normally go before a finite verb in ordinary statements.

So:

• Μού ζήτησαν = They asked me

This is the normal position for clitics in Modern Greek.
You will often see this with other pronouns too:

• μου είπε = he told me
• σου έδωσαν = they gave you
• της ζήτησα = I asked her

Because Greek often uses third person plural with no explicit subject when the subject is indefinite or obvious from context.

So ζήτησαν literally means they asked, but it often works like English they when we do not name the people directly.

Here it means something like:

• At the other branch, they asked me for...

The unnamed they is probably the staff/employees there.

This is a difference of aspect:

• ζήτησαν = aorist → a single completed action
• ζητούσαν = imperfect → ongoing, repeated, or habitual action in the past

So:

• μού ζήτησαν = they asked me
• μού ζητούσαν = they were asking me / they used to ask me

In this sentence, the speaker is referring to one completed request, so ζήτησαν is the natural choice.

Άλλο ένα means another one or one more.

So:

• άλλο ένα δικαιολογητικό = one more supporting document

This is a very common Greek pattern:

• άλλο ένα λεπτό = one more minute
• άλλο ένα ποτήρι = another glass
• άλλο ένα πρόβλημα = one more problem

A useful nuance:

• άλλο δικαιολογητικό can mean a different document
• άλλο ένα δικαιολογητικό clearly means an additional document

Because the sentence lists two separate things they asked for:

1. άλλο ένα δικαιολογητικό
2. ένα αποδεικτικό διεύθυνσης

Repeating ένα keeps the two noun phrases clearly separate. In English we often do the same:

• another document and a proof of address

If the article were not repeated, the structure would feel less clear and less natural.

Because διεύθυνσης is in the genitive, and Greek often uses this kind of noun + genitive structure for categories like proof of X, certificate of Y, application for Z, especially in official or bureaucratic language.

So:

• αποδεικτικό διεύθυνσης = proof of address

Here διεύθυνσης is genitive singular of διεύθυνση.

The version with the article (της διεύθυνσης) would sound more specific, as if referring to a particular address already under discussion. In this sentence, the meaning is more general: a document proving the person's address.

They are both in the accusative singular, because they are the direct objects of ζήτησαν.

• ζήτησαν άλλο ένα δικαιολογητικό
• και ένα αποδεικτικό διεύθυνσης

Both nouns are neuter singular, and in many neuter nouns the nominative and accusative forms are identical, so the form does not change.

That is a good thing to notice. In many modern texts, you will simply see μου without an accent. Some writers also write μού, especially before a verb, to show the stress more clearly or for stylistic/typographic reasons.

So in practice:

• μου ζήτησαν
• μού ζήτησαν

mean the same thing here.

For a learner, the important point is the function: it is the weak pronoun meaning to me / me.