teeburu wo hukin de huita ato, araimono mo kataduketa.

Vた + あと means after doing V.

So:

• ふいた = wiped
• あと = after

Together, ふいたあと means after wiping.

This pattern is very common:

• 食べたあと = after eating
• 帰ったあと = after going home
• 勉強したあと = after studying

In this sentence, it shows that the action of wiping the table happened first, and then the other action happened afterward.

Because あと normally follows the plain past form of the verb when you mean after doing something.

The dictionary form of the verb is ふく (to wipe).
Its plain past form is ふいた.

So:

• ふく = wipe / to wipe
• ふいた = wiped
• ふいたあと = after wiping

This is not the て-form pattern.
If you used ふいて, that would connect clauses in a different way, more like wipe and... rather than clearly after wiping...

Also, ふく → ふいた follows a common sound change for verbs ending in く:

• 書く → 書いた
• 聞く → 聞いた
• ふく → ふいた

を marks the direct object of the verb—the thing directly affected by the action.

Here, the verb is ふく (to wipe), and the thing being wiped is テーブル (table).

So:

• テーブルをふく = wipe the table

This is the normal pattern:

• ドアを開ける = open the door
• 本を読む = read a book
• 皿を洗う = wash dishes / wash a plate

Here, で marks the tool or means used to do the action.

So ふきんでふいた means wiped it with a dishcloth.

This use of で is very common:

• はしで食べる = eat with chopsticks
• ペンで書く = write with a pen
• バスで行く = go by bus

So in this sentence:

• テーブルを = the table
• ふきんで = with a dishcloth
• ふいた = wiped

ふきん means a dishcloth, cleaning cloth, or rag used for wiping.

It comes from the verb ふく (to wipe), so the word has the sense of something for wiping.

You may sometimes see it written in kanji as 布巾, but it is very often written in hiragana because:

• the kanji are not especially common in everyday writing
• hiragana can feel softer and more natural for household vocabulary

So seeing ふきん in hiragana is completely normal.

Yes—that is basically how it is built.

• 洗い comes from 洗う (to wash)
• 物 means thing(s)

So 洗い物 literally means things to wash or washing items, but in natural English it often means:

• the dishes
• the washing up

Depending on context, it can refer to the dirty dishes themselves or the dishwashing task in general.

This kind of noun formation is common in Japanese:

• 買い物 = shopping / things to buy
• 読み物 = reading material
• 編み物 = knitting / knitted things

も means also, too, or as well.

So 洗い物も片付けた implies that, in addition to wiping the table, the speaker also dealt with the dishes.

It adds the idea of another thing was done too.

Compare:

• 洗い物を片付けた = put away / dealt with the dishes
• 洗い物も片付けた = also put away / dealt with the dishes

The も gives a small sense of not only that, but this too.

片付ける is a very useful verb that often means:

• to put away
• to tidy up
• to clear away
• to take care of

In this sentence, with 洗い物 as the object, it most likely means something like:

• put the washed dishes away
• cleared up the dishes
• finished dealing with the dishes

It does not necessarily mean washed the dishes—that idea is already in 洗い物 as the category of thing being dealt with.
片付ける focuses on putting things in order / clearing them away.

Because the sentence is describing a sequence of completed actions.

• ふいた = wiped
• 片付けた = put away / tidied up

Even though English sometimes says After wiping the table, I put away the dishes, Japanese uses the past form before あと:

• ふいたあと = after wiping

Then the main action is also in the past because it also happened and is complete:

• 片付けた = put away / tidied up

So the whole sentence sounds like someone is reporting what they did.

Yes, the subject is omitted.

Japanese often leaves out the subject when it is obvious from context. In a sentence like this, the natural assumption is that the same person did both actions.

So the sentence does not explicitly say I, she, or they, but context usually makes that clear.

This is extremely common in Japanese. A full sentence with a subject would often sound unnecessary unless the speaker specifically wanted to emphasize who did it.

The sentence uses the plain past forms:

• ふいた
• 片付けた

This makes it sound like:

• casual speech
• diary style
• narrative style
• informal written explanation

If you wanted to make it polite, you could say:

• テーブルをふきんでふいたあと、洗い物も片付けました。

Notice that ふいたあと stays the same. The final verb is often where politeness is shown most clearly.

So the original sentence is not wrong or incomplete—it is just in a plain, non-polite style.