I hope you're well. I've copied your letter below with some of my suggestions about how it could be improved. I really hope you get the job!
Feel free to message me should you have any further questions about anything, or if you'd like to discuss the letter in more depth - it might be useful to write a bit more about your previous work experience and mention how that could help you to be a success in the new job.
My name is Yaroslav Dzendziura. I am writing to apply for the position of waiter that I heard about from my friend Zelenskiy Dmitriy, who is studying at the EU Business School in Barcelona, Spain. I have attached my resume to this cover letter for your perusal.
This summer, I will be 18 years old and I am looking for some part time work to show my parents that I can be independent. Moreover, given the fact that I'm studying International Relations, I believe that the role will provide me with a valuable opportunity to improve my English language skills, as I will be able to interact with customers and other staff members on a daily basis.
I strongly believe that I have the experience and the ability to be successful within this position, as I previously worked at a busy restaurant as a waiter (add location > specific details), and whilst doing that job I learned a lot of key skills (perhaps add a few of the skills that you learned - time keeping, proficient customer service, team work etc.)
Due to university commitments and my exam timetable, I will be able to start working in July.
I look forward to hearing from you soon, and please get in touch should you have any questions regarding my application.
(Name > contact details)