The weather atin May promised to be lovely, and so Tom and Kate agreed to go play some tennis thisat/one Sunday. The weather in May promised to be lovely, and so Tom and Kate agreed to go play some tennis that/one Sunday.

The story is in past tense, "this Sunday" refers to upcoming sunday in future.

The weather atin May promised to be lovely, and so Tom and Kate agreed to go play some tennis thison Sunday. The weather in May promised to be lovely, and so Tom and Kate agreed to go play some tennis on Sunday.

You could say "weather in May" or "weather during May", but not "weather at May" "this Sunday" is fine, but "on Sunday" sounds more natural

Kate is always punctual, so she had got toreached Tom's apartment on time. Kate is always punctual, so she had reached Tom's apartment on time.

Kate is always punctual, so she had got to Tom's apartment on time. Kate is always punctual, so she got to Tom's apartment on time.

You could also say "so she had gotten to" or "so she arrived at"

He shared that today he got so excited about playing tennis that he accidentally pulled the racket in different directionsapart, and it broke down. He shared that today he got so excited about playing tennis that he accidentally pulled the racket apart, and it broke.

"pulled the racket in different directions" does not make sense. I am not sure what you mean? "broke down" is usually used for things that break after being used too much, and usually refers to either mechanical things. For example, "After being driven across the country, the car broke down." Sometimes it is also used for people, if they have been through something stressful. For example, "After a stressful exam period, Kate broke down and began to cry."

Kate thought for a moment thought about what happened. Kate thought for a moment about what happened.

Her mother used to say to her that she should always strive to remain optimistic tno matter what happens. Her mother used to say to her that she should always strive to remain optimistic no matter what happens.

And Kate attempted to follow this advisce. And Kate attempted to follow this advice.

"Advise" is a verb, "advice" is a noun. Kate's mother advised her. Kate followed her mother's advice.

She said to Tom, Ddon't worry. She said to Tom, don't worry.

I will buy you a new one as a present for your upcoming birthday! I will buy you a new one as a present for your birthday!

"upcoming birthday" is fine, but does not sound very natural

He thanked her earnestly, and they hurried upoff to the store. He thanked her earnestly, and they hurried off to the store.

There, Kate told Tom he could choose any racket he liked and she would buy it. There, Kate told Tom he could choose any racket he liked and she would buy it.

Excited, Tom started looking over the rackets the store washad on offering. Excited, Tom started looking over the rackets the store had on offer.

Excited, Tom started looking over the rackets the store was offeringhad for sale. Excited, Tom started looking over the rackets the store had for sale.

"was offering" is okay, but "had for sale" is more natural. You can also say "Tom started looking over the rackets the store had for sale excitedly."

There were many allurtempting offers. There were many tempting offers.

alluring is okay, but is usually applied to people: "She was a beautiful, alluring woman."

He was eager to try out athis new racket, and so they hurried to the tennis court. He was eager to try out this new racket, and so they hurried to the tennis court.

SThe sun was shining, and it was the perfect setting for a game of tennis. The sun was shining, and it was the perfect setting for a game of tennis.

Tom got so excited that he played tennis better than usual and beat Kate in the first two matches. Tom got so excited that he played tennis better than usual and beat Kate in the first two matches.

Tom gotwas so excited that he played tennis better than usual, and beat Kate in the first two matches. Tom was so excited that he played tennis better than usual, and beat Kate in the first two matches.