Toplantının gündemini görünce ilk slaytı yeniden hazırladım.

It has two parts:

Toplantının gündemini görünce = when/upon seeing the meeting agenda
ilk slaytı yeniden hazırladım = I prepared the first slide again / I redid the first slide

A very common Turkish pattern is:

[time/circumstance clause] + [main clause]

So the sentence literally builds up the situation first, then gives the main action at the end.

Görünce comes from the verb görmek = to see.

It is formed as:

gör- + -ünce

This ending is the -ince/-ınca/-ünce/-unca form, which usually means when, upon, or sometimes after doing something.

So:

görünce = when I saw / when seeing / upon seeing

In this sentence, it connects the first action to the second one: I saw the agenda, and then I redid the slide.

This is a possessive noun phrase, basically the agenda of the meeting.

It breaks down like this:

toplantı = meeting
toplantının = of the meeting
gündem = agenda
gündemi = its agenda / the agenda of it
gündemini = the agenda of the meeting + accusative case

So:

toplantının gündemi = the meeting’s agenda
toplantının gündemini = the meeting’s agenda as a direct object

This is a very common Turkish pattern: [possessor in genitive] + [possessed noun with possessive ending]

Because it is doing two jobs at once.

First, it belongs to the possessive construction:

toplantının gündemi = the meeting’s agenda

Here, gündemi has the 3rd person possessive ending.

Then that whole noun becomes the direct object of görmek:

Toplantının gündemini görünce = when seeing the meeting agenda

So Turkish adds the accusative on top of the possessive form.

That is why you get:

gündem + i + ni

The word is not “double marked by accident”; each ending has a separate function.

That n is a buffer consonant.

When a noun already has a 3rd person possessive ending and then takes another ending like the accusative, Turkish often inserts n between them.

So:

gündemi + accusative → gündemini

This happens in many words:

araba → arabası → arabasını
ev → evi → evini

So the n is there for pronunciation and grammar; it is a normal part of the pattern.

Because it is a specific direct object.

In Turkish, direct objects are usually marked with the accusative when they are definite or specific.

Here, ilk slayt does not mean just any slide. It means a particular one: the first slide.

So:

ilk slaytı hazırladım = I prepared the first slide

If you leave off the accusative, the meaning becomes less definite, more like I prepared a first slide / some first slide, which does not fit this context well.

Because Turkish often leaves out subject pronouns when the verb already shows who the subject is.

The ending in hazırladım tells you the subject is I:

hazırla- = prepare
-dı = past tense
-m = I

So hazırladım already means I prepared.

You could add ben for emphasis:

Ben ilk slaytı yeniden hazırladım.

But it is not necessary in a neutral sentence.

Yeniden means again, anew, or from scratch again.

In this sentence, it suggests that the speaker reworked or redid the slide after seeing the agenda.

Tekrar can also mean again, and in many contexts they are similar. But with verbs like hazırlamak, yeniden often sounds a bit more like re-prepare / redo / remake, while tekrar can sound more like simple repetition.

So:

yeniden hazırladım = I prepared it again / I redid it

Yes. In English, all of those can be natural translations depending on context.

The Turkish ending -ince/-ınca/-ünce/-unca mainly gives a sense of at the time of doing something or once that happened.

So in this sentence:

Toplantının gündemini görünce...

could be understood as:

When I saw the meeting agenda...
Upon seeing the meeting agenda...
After seeing the meeting agenda...

The Turkish form itself does not strongly emphasize a long time gap. It usually suggests a fairly direct sequence.

Yes, you could say:

Toplantının gündemini gördüğümde ilk slaytı yeniden hazırladım.

That is grammatical.

The difference is mostly in style and nuance:

görünce = more compact, very common for when/once I saw it
gördüğümde = more literally when I saw it / at the time I saw it

In a sentence like this, görünce sounds very natural because it describes one action leading into another.

It is the simple past (also often called the definite past in Turkish grammar).

Breakdown:

hazırla- = prepare
-dı = past
-m = I

So:

hazırladım = I prepared

Depending on context, English might translate it as:

I prepared
I redid
I revised

because the exact English verb depends on the situation.

Because Turkish normally places the finite verb at the end of the clause.

So instead of English-style order like:

I redid the first slide after seeing the meeting agenda

Turkish naturally prefers:

After seeing the meeting agenda, I the first slide again prepared

Of course that word-for-word English is unnatural, but it shows the Turkish logic: important grammatical information often comes at the end, especially the main verb.

Because slayt is a loanword, and in standard Turkish it keeps its t before vowel-initial endings:

slayt → slaytı

In many native Turkish words, final consonants like p, ç, t, k may soften before a vowel:

kitap → kitabı

But many borrowed words do not behave the same way. Slayt usually stays slayt-.

So ilk slaytı is the expected form.

Yes, it is very natural.

It sounds like something a person might say in a work or presentation context: they saw the meeting agenda and then decided to redo the first slide.

A slightly freer English rendering would be:

After seeing the meeting agenda, I redid the first slide.

That captures the natural flow well, and the Turkish sentence itself is perfectly normal.