My class has entered a competition and we hope we will win first prize.
You could also say:
My class has entered a competition and we hope to win first prize.
Some may even suggest that you use a comma after "competition" to provide a short pause between the two thoughts. It is not obligatory.
My class has entered a competition, and we hope we will win first prize.
My class has entered a competition, and we hope to win first prize.
John

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