Avia puellam vocat, ut secum lunam per fenestram spectet.

It has two parts:

• Avia puellam vocat = the main clause
• ut secum lunam per fenestram spectet = a subordinate clause introduced by ut

In the main clause, avia is the subject, puellam is the direct object, and vocat is the verb.
The ut clause explains the purpose: why the grandmother calls the girl.

Because puellam is the direct object of vocat.

• avia = the grandmother (subject, nominative)
• puellam = the girl (direct object, accusative)
• vocat = calls

So the grandmother is doing the calling, and the girl is the one being called.

They are accusative for different reasons.

• lunam is the direct object of spectet: she watches the moon
• fenestram is accusative because it follows per, and per takes the accusative

So:

• lunam = object of the verb
• per fenestram = prepositional phrase, through the window

Even though both nouns are accusative, the preposition per makes fenestram easy to identify.

Here ut introduces a purpose clause. It means something like:

• so that
• in order that

So ut ... spectet means so that she may watch or in order to watch.

This is a very common Latin pattern: ut + subjunctive to express purpose.

Because it is inside a purpose clause.

Latin normally uses:

• ut + subjunctive for purpose

So spectet is not saying she is watching as a plain fact.
It is expressing the intended result or aim of the calling: the grandmother calls the girl so that she may watch.

Also, since the main verb is present (vocat), Latin uses the present subjunctive (spectet) here.

The understood subject is most naturally the girl.

Latin often leaves out subject pronouns when the meaning is clear from context. In this sentence, the idea is:

• the grandmother calls the girl
• so that the girl may watch the moon

So even though no separate word for she appears, spectet is understood with puella as its subject.

Secum means with her/him/self.

In Latin, with personal and reflexive pronouns, cum is usually attached to the end:

• mecum = with me
• tecum = with you
• secum = with herself / himself / with her / him (reflexive)
• nobiscum = with us
• vobiscum = with you all

So secum is just the normal form instead of cum se.

In this sentence, secum most naturally refers back to avia, the grandmother: with her.

This is a point that often surprises learners.

In subordinate clauses, a reflexive like se / secum can refer back to the subject of the main clause, especially when the subordinate clause is closely connected to that subject's intention or purpose.

So here the sense is:

• The grandmother calls the girl
• so that the girl may watch the moon with the grandmother

That is why secum is understood as with her = with the grandmother, not with herself.

Because Latin has no articles.

So:

• avia can mean grandmother, a grandmother, or the grandmother
• puellam can mean a girl or the girl
• lunam usually becomes the moon from context

English requires articles, but Latin usually leaves that idea to context.

Latin word order is much more flexible than English word order because the endings show each word's role.

This order is perfectly natural:

• Avia puellam vocat first gives the main action
• then ut secum lunam per fenestram spectet gives the purpose

A different order could still make sense if the case endings stayed the same. Latin word order often helps with emphasis more than with basic grammar.