На чашке есть потёртость.

Чашке is in the prepositional case because it follows на in a location meaning: on the cup.

Base form: чашка
Prepositional singular: на чашке

So the pattern is:

• на столе — on the table
• на двери — on the door
• на чашке — on the cup

Here, Russian treats the worn mark as being on the surface of the cup.

Because the damage is on the outside surface of the cup, not inside it.

• на чашке = on the cup / on the surface of the cup
• в чашке = in the cup / inside the cup

A потёртость is a mark on the surface, so на is the natural choice.

Here есть means there is.

So the structure is basically:

• На чашке есть потёртость.
• There is a worn/scuffed spot on the cup.

Russian often uses есть when introducing the existence of something. In this sentence, it helps present потёртость as something that exists on the cup.

Yes, you can, and Russian speakers do this.
But there is a slight difference in feel:

• На чашке есть потёртость. — neutral, full there is statement
• На чашке потёртость. — shorter, more matter-of-fact, like noticing or pointing it out

Both are understandable and natural. The version with есть sounds a bit more explicit.

Потёртость is a worn spot, scuff, or rubbed mark caused by friction or use.

It is not exactly the same as:

• царапина — a scratch
• трещина — a crack
• пятно — a stain

So if the cup has a place where the surface or paint has been rubbed down, потёртость is a very good word.

Because nouns ending in -ость are usually feminine in Russian.

For example:

• скорость — speed
• молодость — youth
• твёрдость — hardness
• потёртость — worn spot / scuffing

So in this sentence, потёртость is feminine singular nominative.

The most neutral word order is:

• На чашке есть потёртость.

This starts with the location and then introduces what exists there.

Other orders are possible, but they change the emphasis:

• Потёртость есть на чашке. — unusual, strongly emphasizes the worn mark
• На чашке потёртость. — shorter, more direct
• Потёртость на чашке. — can work as a label or brief observation, but less like a full neutral sentence

So for a learner, На чашке есть потёртость is the safest standard version.

It is much less natural.

У X есть Y usually means X has Y in a possession-like sense:

• У меня есть книга. — I have a book.

But with physical marks on an object, Russian normally uses the location on the surface:

• На чашке есть потёртость.

Using у чашки есть потёртость sounds like the cup possesses the flaw, which is grammatically understandable but not the usual way to say it.

Yes, ё matters a lot. It is always stressed.

• потёртость = stress on тёр
• roughly: pa-TYOR-tast'

A more careful approximation:

• по sounds reduced
• тёр is the stressed part
• the final -сть is soft

Also:

• чашке is stressed on the first syllable: ЧАШ-ке

So the whole sentence is approximately:

на ЧАШ-ке есть па-ТЁР-тасть

Usually чашка means cup, especially a teacup or a cup used with a saucer.
A mug is more often кружка.

Still, in casual speech the boundary is not always strict, and context matters. If the object is clearly a mug, many speakers would prefer:

• На кружке есть потёртость.

But if your source gives чашка, then the sentence is perfectly fine for cup.

You would use the plural:

• На чашке есть потёртости. — There are worn spots on the cup.

Or, more naturally in many cases:

• На чашке потёртости.

So the noun changes from singular потёртость to plural потёртости. The rest of the sentence can stay the same.