Der Stich an meiner Hand juckt schon wieder.

In this sentence, der Stich most naturally means a sting or bite mark, for example from a mosquito or another insect.

German Stich can also mean:

• a stab or prick
• a stitch in sewing
• a sudden pain, depending on context

Here, because it itches and is on the hand, learners will usually understand it as an insect bite/sting.

Stich is a masculine noun, so its nominative singular article is der.

Here it is the subject of the sentence:

• Der Stich = the bite/sting
• juckt = it itches

So:

• der = masculine nominative singular

Unfortunately, noun gender usually has to be learned along with the noun:

• der Stich
• die Hand

German often uses an for something located on or attached to a body part.

So:

• an meiner Hand = on my hand / on my hand area

Why not the others?

• auf meiner Hand would sound more like something is lying on top of your hand
• in meiner Hand means in my hand, as in inside my hand or in my grasp

For a bite/sting located on a body part, an is very natural:

• ein Kratzer an meinem Arm
• eine Wunde an seinem Bein

Because an here takes the dative case.

This is a location, not a direction:

• an meiner Hand = at/on my hand

With two-way prepositions like an, German uses:

• dative for location
• accusative for movement toward somewhere

So:

• an meiner Hand = on my hand, at my hand → dative
• an meine Hand would suggest movement toward the hand, which does not fit here

Also, Hand is feminine:

• nominative: meine Hand
• dative: meiner Hand

Because this sentence describes a fixed location: where the sting is.

The preposition an is one of the two-way prepositions. These can take either:

• dative for location
• accusative for direction/movement

Compare:

• Der Stich ist an meiner Hand. = The sting is on my hand. → location, so dative
• Ich halte die Flasche an die Wand. = I hold the bottle against the wall. → movement toward the wall, so accusative

In your sentence, there is no movement. The sting is simply located there.

Because the subject is singular: der Stich.

The structure is:

• Der Stich = singular subject
• juckt = 3rd person singular of jucken

So the verb agrees with Stich, not with Hand.

Compare:

• Der Stich juckt. = The bite itches.
• Die Stiche jucken. = The bites itch.

Yes, very similarly.

jucken means to itch:

• Der Stich juckt. = The bite itches.
• Meine Haut juckt. = My skin itches.

You can also use it with a person as the affected object:

• Die Hand juckt mich. = My hand itches / My hand is itchy
  This structure exists, but in everyday German people often simply say:
• Meine Hand juckt.

So in your sentence, the thing doing the itching is der Stich.

Schon wieder means again already, but in natural English it is usually just again with a feeling of annoyance, surprise, or repetition.

So:

• Der Stich an meiner Hand juckt schon wieder. = The bite on my hand is itching again.

The nuance is often something like:

• it had stopped
• and now it has started again
• possibly annoyingly

Compare:

• wieder = again
• schon wieder = again already / yet again

Both can mean again, but schon wieder often sounds more like:

• again already
• yet again
• once again, often with mild irritation

Examples:

• Es regnet wieder. = It’s raining again.
• Es regnet schon wieder. = It’s raining again already / It’s raining yet again.

In your sentence, schon wieder suggests the itching has returned, probably in an annoying way.

This is normal German main-clause word order:

• Der Stich an meiner Hand = subject phrase
• juckt = finite verb in position 2
• schon wieder = adverbial expression later in the clause

German main clauses usually place the finite verb in the second position:

• Der Stich an meiner Hand | juckt | schon wieder

The whole phrase Der Stich an meiner Hand counts as one unit before the verb.

You could also change the emphasis:

• Schon wieder juckt der Stich an meiner Hand.

That is possible, but less neutral.

It mainly describes Stich: it tells you which sting/bite we are talking about.

So the phrase is best understood as:

• [Der Stich an meiner Hand] = the sting on my hand
• juckt schon wieder = is itching again

In other words, an meiner Hand is part of the noun phrase, not just a random location added to the verb.

It is possible in some contexts, but it would usually mean something different.

• an meiner Hand = on my hand / at my hand
• in meiner Hand = in my hand

For an insect bite on the surface of your skin, an meiner Hand is much more natural.

In meiner Hand might suggest:

• something inside the hand
• a pain felt within the hand
• or that you are holding something in your hand

So for a bite/sting mark on the skin, an is the better choice.

Sometimes German uses the definite article with body parts, especially when the owner is clear from context:

• Ich habe mir die Hand verletzt.

But in this sentence, an meiner Hand is perfectly natural and often clearer.

Compare:

• Der Stich an der Hand juckt. = The bite on the hand itches.
  This can sound more general or less personal.
• Der Stich an meiner Hand juckt. = The bite on my hand itches.
  This clearly identifies whose hand it is.

So yes, German often uses articles with body parts, but here the possessive meiner is completely normal.

It could be several things. Stich is not limited to mosquitoes.

Depending on context, it could mean:

• a mosquito bite
• a bee or wasp sting
• another insect sting/bite
• a prick or puncture that now itches

If a speaker wanted to be more specific, they might say:

• Mückenstich = mosquito bite
• Bienenstich = bee sting
• Wespenstich = wasp sting

But der Stich by itself is often enough when the situation is obvious.