Ψάχνω καφέ που να έχει σκιερά τραπέζια και λίγο αεράκι το απόγευμα.

Here καφέ means café, not coffee. As a noun meaning café, it is commonly treated as an indeclinable neuter noun: το καφέ.

So in this sentence, καφέ stays the same in form.

Also, after a verb like ψάχνω, Greek often leaves out the indefinite article when the meaning is general:

Ψάχνω καφέ = I’m looking for a café
Ψάχνω ένα καφέ = I’m looking for a café / one café

Both are possible, but the version without the article can sound a bit more general or natural in this kind of context.

One thing that can confuse learners is that καφέ is also the accusative form of ο καφές = coffee. So context is what tells you which meaning is intended.

Because the Greek present tense covers both:

• I look for
• I am looking for

So Ψάχνω can mean either, depending on context. Greek does not need a separate progressive form here the way English often does.

That makes Ψάχνω καφέ... a perfectly normal way to say I’m looking for a café...

This is one of the most important points in the sentence.

που να έχει is used because the speaker is talking about a desired kind of café, not a specific café already known to exist.

So:

• που έχει = that has / which has
  This sounds more factual, as if you have a real café in mind.
• που να έχει = that would have / that should have
  This sounds like a requirement, wish, or desired characteristic.

After verbs like ψάχνω, θέλω, χρειάζομαι, Greek very often uses this pattern:

noun + που να + verb

Examples:

• Ψάχνω σπίτι που να είναι ήσυχο.
• Θέλω δουλειά που να πληρώνει καλά.

So in your sentence, που να έχει means something like that has / that would have in the sense of a café meeting these conditions.

που introduces a relative clause, like English that or which.

So:

• καφέ που... = café that...

In everyday Greek, που is the most common way to do this. It does not change form for gender, number, or case the way a more formal relative pronoun might.

Here it links καφέ with the description that follows:

καφέ που να έχει σκιερά τραπέζια...
= a café that would have shaded tables...

Because the subject of έχει is καφέ, not τραπέζια.

The structure is:

• the café has tables

So the verb agrees with καφέ, which is singular.

That is why Greek uses:

που να έχει σκιερά τραπέζια

not που να έχουν...

Even though τραπέζια is plural, it is the object of έχει, not the subject.

Because σκιερά agrees with τραπέζια.

τραπέζια is:

• neuter
• plural

So the adjective must also be neuter plural:

• masculine singular: σκιερός
• feminine singular: σκιερή
• neuter singular: σκιερό
• neuter plural: σκιερά

So:

• σκιερό τραπέζι = shaded table
• σκιερά τραπέζια = shaded tables

This is standard adjective agreement in Greek.

It means more than just a little wind in a neutral sense.

αεράκι is a diminutive form, and it usually suggests a light, pleasant breeze. So it sounds softer and nicer than a plain word for wind would.

And λίγο adds the idea of a bit of.

So λίγο αεράκι gives a very natural, almost cozy image: not strong wind, just a gentle breeze.

Grammatically, λίγο is in the neuter singular form, matching the neuter noun αεράκι.

Because Greek often uses the accusative by itself for time expressions.

So:

• το πρωί = in the morning
• το μεσημέρι = at noon / in the midday period
• το απόγευμα = in the afternoon
• το βράδυ = in the evening / at night

You do not need a preposition like in here. Greek simply says το απόγευμα.

In this sentence it tells you when the speaker wants that breeze: ...λίγο αεράκι το απόγευμα = ...a little breeze in the afternoon.

Yes, you could, and it sounds natural.

Using με gives a more compact description:

• καφέ με σκιερά τραπέζια... = a café with shaded tables...

That works especially well for concrete features.

But που να έχει emphasizes the idea of the kind of place I want to find, with desired qualities. It can sound a little more search-oriented or requirement-oriented.

So both are good, but they are slightly different in feel:

• με = with
• που να έχει = that would have / that should have

The word order is fairly flexible.

Your sentence is natural as it stands, but Greek could move parts around for emphasis. For example:

Ψάχνω καφέ που να έχει λίγο αεράκι το απόγευμα και σκιερά τραπέζια.

That is still grammatical, though the emphasis shifts a bit.

The original order sounds very natural because it presents the desired features in a smooth list:

1. σκιερά τραπέζια
2. λίγο αεράκι
3. το απόγευμα

So while Greek allows movement, the version you have is a good neutral, idiomatic one.