Ο παίκτης περιμένει τη σέντρα και μετά κλωτσάει δυνατά.

Because the Greek verb περιμένω normally takes a direct object.

So Greek says:

περιμένω το λεωφορείο = I wait for the bus
περιμένει τη σέντρα = he waits for the cross/kickoff

In English, wait usually needs for, but in Greek περιμένω does not.

They are in different cases because they do different jobs in the sentence.

• ο παίκτης = the subject, so it is in the nominative
• τη σέντρα = the direct object, so it is in the accusative

The article changes to show this:

• masculine nominative singular: ο
• feminine accusative singular: τη / την

Also, παίκτης is masculine, while σέντρα is feminine.

Many feminine nouns ending in -α have the same form in the nominative and accusative singular.

So:

• η σέντρα = the center/cross/kickoff as a subject
• τη σέντρα = the center/cross/kickoff as an object

The article shows the case more clearly than the noun ending does.

The full feminine accusative article is την, but the final -ν is often dropped before certain consonants.

Since σέντρα begins with σ, τη σέντρα is the normal form here.

So:

• την
  • vowel or certain consonants: the -ν is usually kept
• before σ: it is very often dropped

That is why τη σέντρα sounds natural.

Because ο παίκτης means the player, a specific player.

Greek uses the definite article very naturally with specific nouns, often more regularly than English learners expect.

Compare:

• ο παίκτης = the player
• ένας παίκτης = a player

A bare noun like παίκτης can appear in special contexts, but in a normal sentence like this, ο παίκτης is the expected form.

Περιμένει is:

• present tense
• 3rd person singular
• from the verb περιμένω

So it means:

• he waits
• she waits
• it waits
• or, depending on context, he/she is waiting

Greek present tense can cover both a simple present and a present-in-progress meaning.

Κλωτσάει is also:

• present tense
• 3rd person singular
• from the verb κλωτσάω

So it means he/she kicks or is kicking.

A useful extra point: with verbs like κλωτσάω, Greek often has two common present forms in some persons:

• κλωτσάει
• κλωτσά

Both can mean he/she kicks.
Κλωτσάει is the fuller form and is very common in teaching materials and everyday Greek.

Because Greek often leaves out subject pronouns when they are not needed.

The verb ending already tells you the person, and here the subject is already named:

Ο παίκτης περιμένει ...

So adding αυτός would usually be unnecessary unless you wanted special emphasis, such as he, not someone else.

Here και μετά means and then or and afterwards.

In this sentence:

• και = and
• μετά = then / afterwards

So it links the two actions in sequence:

1. he waits
2. then he kicks

Also, μετά can be a preposition meaning after, for example:

μετά το παιχνίδι = after the game

But in your sentence, it is an adverb, meaning then.

Because δυνατά is an adverb, and it describes how he kicks.

• κλωτσάει δυνατά = he kicks hard / strongly

By contrast:

• δυνατός is an adjective for a masculine noun
• δυνατή is an adjective for a feminine noun

For example:

• δυνατός παίκτης = a strong player

So:

• δυνατός describes a noun
• δυνατά describes a verb

Not completely. Greek word order is more flexible than English word order.

The sentence as given:

Ο παίκτης περιμένει τη σέντρα και μετά κλωτσάει δυνατά

is a very neutral, natural order.

Greek can move things around for emphasis, contrast, or style, but the basic version here is the best one for a straightforward statement.

So for a learner, this order is a good model: subject + verb + object + and then + verb + adverb

A few useful points:

• αι in παίκτης sounds like e in met
• ει in περιμένει sounds like ee in see
• δ in δυνατά sounds like th in this
• τσ in κλωτσάει sounds like ts in cats

A rough pronunciation guide:

o PEK-tis pe-ri-ME-ni ti SEN-tra ke me-TA klo-TSA-i thi-na-TA

This is only an approximation, but it will help you get close.