Jó a kapcsolatom az énekessel.

Because Hungarian usually marks possession directly on the noun.

• kapcsolat = relationship / connection
• kapcsolatom = my relationship / connection

The ending -om means my here. So Hungarian does not need a separate word like my unless you want extra emphasis: az én kapcsolatom.

That is normal in Hungarian. A possessed noun often still takes the definite article.

So:

• a kapcsolatom = my relationship
• literally something like the my-relationship, though you should not translate it word for word

In this sentence, a kapcsolatom sounds natural and standard. Leaving out a would usually sound incomplete or would belong to a different structure.

In Hungarian, the verb to be is usually omitted in the present tense, 3rd person when the predicate is a noun, adjective, or similar expression.

So:

• Jó a kapcsolatom az énekessel. = My relationship with the singer is good.

There is no van here, because Hungarian normally leaves it out in this type of sentence.

Compare:

• A kapcsolatom jó. = My relationship is good.
• Én jó vagyok. = I am good.
  Here vagyok is needed because it is 1st person, not 3rd person.

It is énekes + the ending -val/-vel, which means with.

• énekes = singer
• énekessel = with the singer

The full phrase:

• az énekessel = with the singer

So the sentence is built around my relationship with the singer.

This is because of assimilation, which is very common in Hungarian.

The suffix is basically -val/-vel, but when it is attached to a word ending in a consonant, the v often disappears and the following consonant doubles.

So:

• énekes + vel → énekessel

The final s of énekes becomes doubled in pronunciation and spelling: ss.

You see the same pattern in other words too:

• Péterrel = with Péter
• kézzel = with a hand
• vonattal = by train / with a train

Because Hungarian has two forms of the definite article:

• a before a consonant sound
• az before a vowel sound

Since énekes begins with a vowel, Hungarian uses az:

• az énekes
• az énekessel

This is similar in principle to English a vs an.

Hungarian word order is flexible and often reflects focus or what is being presented first.

Starting with Jó puts the quality good in a prominent position:

• Jó a kapcsolatom az énekessel.

This is very natural, especially if the important point is that the relationship is good.

Other word orders are possible, but they shift emphasis:

• Az énekessel jó a kapcsolatom. = It’s with the singer that my relationship is good.
• A kapcsolatom jó az énekessel. = grammatical, but less natural in many contexts

So the sentence-final and sentence-initial positions are often about emphasis, not just rigid grammar rules.

Not necessarily. Kapcsolat is a broad word. It can mean:

• relationship
• connection
• contact
• rapport

So Jó a kapcsolatom az énekessel usually means something like:

• I have a good relationship with the singer.
• I get on well with the singer.
• I’m on good terms with the singer.

It does not automatically imply romance.

You could, but it does not mean the same thing.

• Jó a kapcsolatom az énekessel. = My relationship with the singer is good.
• Jó a kapcsolat az énekessel. = The relationship with the singer is good / relations with the singer are good

The first sentence is personal and specifically about my relationship. The possessive ending -om is important.

Yes. Kapcsolatom is singular: my relationship / my connection.

Hungarian often uses the singular where English also would, but the important thing is that the noun itself is singular and the possessive ending matches that:

• kapcsolatom = my relationship
• kapcsolataim = my relationships

Here singular is the natural choice.