(I'm learning a contracted formions. If there's are any words that can be contracted, please correct it.)that I have missed, please do so.)¶
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There're some cafes I use to study English after work.

Yesterday, I found out that the cafe I like the most was closed.

To me, this cafe' is a place with something like a spiritual energy,so: my TOIEC score got higher and higher as I studied at this cafthere.

In this case, not using a contraction would sound better.

I went to another second cafe that I like the second, but it was also closed.

That cafe's bread wae bread from that cafe is cheap but tasty.

In this case, not using a contraction would sound better.

So I went to a cafe chainhain-cafe reluctantly.

Minor style point.

CThe coffee and sandwiches' are not good but notor cheap; t. The couple on the right of me started to have a quarrel; t. The two young girls behind me were speaking ill of someone; a. A man and a woman on the front-left were also bad-mouthing someone, s. So I felt helpless.

The limit of my patience toI could only stay there was 15 minutes.

This version is more to the point.

(I'm learning a contracted formions. If there's are any words that can be contracted, please correct ithem.) There' are some cafes I usego to to study English after work.

Typically, you only contract "are" with pronouns, such as "you're" or "we're." I would never contract "there" and "are" when writing.

Yesterday, I found the cafe I like the mostat my favorite cafe was closed.

This sounds more natural.

To me, this cafe's a place with something like spiritual energy, so I felt like my TOIEC score got higher and higher as I studied at this cafe.

I went to another cafe, which I like the second best, but it was also closed.

That cafe's bread was cheap, but tasty.

Reluctantly, I went to a cafe chain reluctantly.

Your sentence is not wrong, but this sounds more natural.

CThe coffee and sandwiches' there are not good, but also not cheap; t. The couple ton the right of me started to have a quarrel; tfight. The two young girls behind me were speaking ill ofbadly about someone; a. A man and a woman on the front-left were also bad-mouthing someone, so I felt helpless.

My changes sound a bit more natural. Things like "quarrel" or "speaking ill" might be considered old-fashioned.

The limiI ran out of my patience to stayafter being there wasfor 15 minutes.