My aim is to improve my English a lot - —both wiritten and spoken.

Note: The type of dash you're looking for there is called an "em dash." Although, it's fine to use a hyphen informally

I am going to write down some random notes from my day-to-day life, rather thean essays.

English has like 100 dumb comma rules. I had to look this one up even. Anyway point is that we don't use a comma before rather than

Maybe it is becaused due poor sleep today - she didn't sleep well today, but my girlfriend hadwas coughing at night.

Today is mythe second day of my English diary.

I'm not sure about this one. Could be correct and I'm just trigger happy

It is quarter past ten and I have not done any of tasks I mentioned above yet :-( I only solve problems, one after another. ¶
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Now there is a calm moment. I will have a cup of coffee and try to finish some emails. ¶
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It was quite a demanding work-day today. Unfortunatelly, I did not managed to do what I wanted to do. I hope tomorrow I will finally manage to reduce my backlog. ¶
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Wednesday, April 13th 2022 ¶
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The weather looks nice today. In the evening I am going to a pub with my friends. I hompe it will be warm in the evening so we couldan sit on a terrace.

I am going to write down some random notes from my day-to-day life, rather thean essays.

'rather' is followed by 'than', not 'then' !

Today I have four appointments, three of them with my colleagues. and one with (...)

personally, this is the structure i would use ! nothing was wrong with your phrasing though, good job !

I have to discuss some important matters concernregarding our projects with them.

concerning is good too, but personally i'd say regarding!

Maybe it is caused due poor sleep todaywas because I did not get enough sleep last night - my girlfriend had a cough at night.

the grammer is a little wrong, and this is a simpler way to phrase it!

I tryam trying to make it a habit of it.

'trying' - present tense because i believe you are still trying (right? haha)

It is quarter past ten (or you could just say 10.45pm) and I have not done any of tasks I mentioned above yet :-( I can (? not sure what you were trying to say here 😥) only solve problems, one after another. ¶
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Now there is a calm moment. I will have a cup of coffee and try to finish some emails. ¶
¶
It was quite a demanding work-day today. Unfortunatelly I did not managed to do what I wanted to do. I hope tomorrow I will finally manage to reduce my backlog. (workload? not sure what a backlog is)¶
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Wednesday, April 13th 2022 ¶
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The weather looks nice today. In the evening I am going to a pub with my friends. I home it will be warm in the evening so we could sit on a terrace.

My aim is to improve my English a lot - both wiritten and spoken.

I am going to write down some random notes from my day-to-day life, rather thean essays.

Maybe it is caused due to poor sleep today - my girlfriend had cough alast night.

I'm trying to make a habit of it.

It is quarter past ten and I have not done any of the tasks I mentioned above yet. :-( I have only solved problems, one after another. ¶
¶
Now there is a calm moment. I will have a cup of coffee and try to finish some emails. ¶
¶
It was quite a demanding work- day today. Unfortunatelly I did not managed to do what I wanted to do. I hope tomorrow I will finally manage to reduce my backlog. ¶
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¶
Wednesday, April 13th 2022 ¶
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The weather looks nice today. In the evening I am going to a pub with my friends. I hompe it will be warm in the evening so we couldan sit on athe terrace.