Проверьте, пожалуйста.
The young officer Almazov wants to go to the academy and to serve in the regiment. Professor at the Academy refuses to accept a job, which was written by the officer. Almazov made a mistake. He painted a bush.
Officer's wife found out about the situation and helped her husband. Vera has sold all her jewelry and she gave the money to the gardener. He planted a lilac bush. Almazov showed bush professor, an officer entered the academy

Не везде понятен смысл, но попробую догадаться. Я бы внёс такие правки:Young officer Almazov wanted to enter the Academy and to serve in the regiment. Professor at the Academy rejected job (painting, может быть, если подходит по смыслу? речь идёт не о написанном сочинении, а о написанной картине? если так, то written не подойдёт) by the officer. Almazov made a mistake - he painted a bush absent in reality. Officer's wife found out about the situation and helped her husband. Vera sold all her jewelry and paid the gardener who planted a lilac bush. Almazov showed professor the bush, an so the officer managed to enter the Academy.