De kapper zegt dat een korte pony handig is in de zomer.

Right — here pony is a hair word, not the animal.

In Dutch, een pony can mean:

• a pony (the animal), or
• a fringe / bangs

Because the sentence starts with De kapper and talks about summer, the hair meaning is clearly intended.

So een korte pony means short bangs or a short fringe.

Because those adjectives are used in two different ways.

• korte is placed before the noun: een korte pony
• handig comes after is: een korte pony is handig

Rule:

• An adjective before a noun usually gets -e:
  een korte pony
• An adjective after zijn is a predicate adjective, and it normally stays in the basic form:
  De pony is handig

A useful comparison:

• een korte pony
• De pony is kort

Good question. After een, adjectives usually still take -e.

The main exception is with a singular het-word:

• een kort kapsel
• een klein huis

But pony is a de-word, so you get:

• een korte pony

So the form korte is exactly what you should expect here.

dat introduces a subordinate clause, like English that in says that...

So the sentence has two parts:

• Main clause: De kapper zegt
• Subordinate clause: dat een korte pony handig is in de zomer

After verbs like zeggen, denken, weten, and horen, Dutch often uses dat to introduce what is being said, thought, known, or heard.

Because after dat, Dutch uses subordinate-clause word order.

In a main clause, the finite verb is usually in second position:

• De kapper zegt ...

In a subordinate clause introduced by dat, the finite verb moves toward the end:

• ... dat een korte pony handig is

So is does not stay in the same place it would in English.

This is one of the biggest word-order differences between English and Dutch.

Because in real Dutch, the verb in a subordinate clause is often near the end, but not always the absolute last word.

You could also say:

• De kapper zegt dat een korte pony in de zomer handig is.

That version may feel more “textbook-neutral” to learners.

The version you were given,

• ... dat een korte pony handig is in de zomer, is also possible and natural. Dutch allows some adverbials or prepositional phrases, such as in de zomer, to appear after the verb.

So the important rule is:

• the verb goes toward the end
• but not necessarily as the final word every single time

Because de and een are doing different jobs.

• de = the
• een = a/an

So:

• de kapper = the hairdresser / barber
• de zomer = the summer
• een korte pony = a short fringe / short bangs

Also:

• kapper is a de-word
• zomer is a de-word
• pony is also a de-word, but you do not see de here because the sentence uses the indefinite article een

kapper can often be translated as hairdresser or barber, depending on context.

A few notes:

• Traditionally, kapper is male and kapster is female.
• In everyday usage, kapper is also often used more generally for the profession or the salon context.

So in this sentence, the safest idea is simply:

• the hairdresser says ...

Because in Dutch, a comma before a dat-clause like this is usually not required.

So this is perfectly normal:

• De kapper zegt dat een korte pony handig is in de zomer.

A writer might sometimes add punctuation for readability in a longer or more complicated sentence, but here no comma is the standard, natural choice.