Badanie wzroku pokaże wadę wzroku i zmierzy rozstaw źrenic.

Wzroku is the genitive singular form of wzrok (eyesight / vision).

It appears twice because Polish often uses the genitive after another noun to show a relationship like of in English:

• badanie wzroku = examination of vision / eye exam
• wada wzroku = defect of vision / vision defect

So even though the English translations may be different, the grammar is the same: noun + genitive noun.

Because wadę wzroku is the direct object of pokaże (will show).

The dictionary form is wada.
But feminine nouns ending in -a usually change to -ę in the accusative singular:

• nominative: wada
• accusative: wadę

So:

• wada wzroku = a vision defect / the vision defect
• pokaże wadę wzroku = will show the vision defect

Because źrenic is the genitive plural of źrenica (pupil of the eye).

The phrase rozstaw źrenic means distance/spacing between the pupils.
Again, Polish uses a noun + genitive structure:

• rozstaw = spacing / distance apart
• źrenic = of the pupils

So rozstaw źrenic literally means something like spacing of the pupils.

They are future in meaning.

Both pokaże and zmierzy are perfective verbs in the non-past form. In Polish, perfective verbs do not normally have a present tense meaning; their non-past forms usually refer to the future.

So:

• pokaże = will show
• zmierzy = will measure

This is very common in Polish. A form that looks “present-like” to an English speaker often translates as future when the verb is perfective.

The subject is badanie wzroku.

That whole phrase is treated as one thing: the eye exam / the eye test.
It is the thing that:

• pokaże wadę wzroku
• zmierzy rozstaw źrenic

So the structure is:

• Badanie wzroku = subject
• pokaże = first verb
• wadę wzroku = first object
• i = and
• zmierzy = second verb
• rozstaw źrenic = second object

Because Polish usually does not need an extra subject pronoun when the subject is already stated.

In English, you might say:

• The eye exam will show... and it will measure...

In Polish, once badanie wzroku has been given as the subject, you can simply continue with the second verb:

• Badanie wzroku pokaże... i zmierzy...

The second verb is still understood to have the same subject.

Polish often leaves out possessive words like your when the meaning is obvious from context.

In a sentence about a medical or optical exam, it is naturally understood that the exam concerns your eyes unless the context says otherwise.

So Polish prefers the shorter form:

• wadę wzroku
• rozstaw źrenic

rather than explicitly saying:

• twoją wadę wzroku
• twój rozstaw źrenic or a more natural rewording with twoich źrenic

Adding your is possible in some contexts, but it is often unnecessary.

Badanie is a verbal noun formed from the verb badać (to examine / to investigate).

It functions like an ordinary noun and means something like:

• examination
• test
• exam

It is neuter singular, which is why you get:

• to badanie = this examination

In the sentence, badanie wzroku is simply a noun phrase meaning eye examination / eye test.

Yes, Polish word order is more flexible than English word order, although not completely free.

The version in your sentence is natural and straightforward:

• Badanie wzroku pokaże wadę wzroku i zmierzy rozstaw źrenic.

You could also rearrange parts for emphasis, for example:

• Badanie wzroku zmierzy rozstaw źrenic i pokaże wadę wzroku.

That still works, but the emphasis changes slightly.
For learners, the original order is a good neutral pattern: subject + verb + object + and + verb + object.

A rough English-friendly guide:

• badanie ≈ ba-DA-nye
• wzroku ≈ VZRO-koo
• wadę ≈ VA-deh (with a nasalized final vowel in careful pronunciation, though in normal speech learners often hear something close to -e)
• rozstaw ≈ ROSS-tahf
• źrenic ≈ ZHRE-nits, but with a very soft Polish ź sound, softer than ordinary English zh

A few notes:

• w in Polish sounds like English v
• rz usually sounds like zh
• ź is a softer sound than rz
• stress in Polish usually falls on the second-to-last syllable

So:

• baDAnie
• WZROku
• waDĘ
• ROZstaw
• ŹREnic

Yes. In optics, another common term is odległość między źrenicami (distance between the pupils) or technical expressions based on PD in some contexts.

But rozstaw źrenic is perfectly normal and concise. It sounds like the kind of wording you might see in an optician’s or medical context.

So in this sentence, zmierzy rozstaw źrenic is a natural way to say that the exam will measure pupil distance.