"The teacher gave expressions/expression to his hope for a new subject." It's an awkward sentence. If it came from your exercise book, I question whether it was written by a native English speaker (though I've seen worse examples in some British and American textbooks). "To give expression to" means to express/make known a thought or feeling. Usually, we give expression to our own thoughts or feelings. I'm not sure how a teacher would express someone else's hope. And for a new subject? It's an odd idea. https://www.merriam-webster.com/dictionary/give%20expression%20to

expression/ = The teacher's actions of some type show that the student has a good reason to hope for a new subject. The teacher have the student a A on the paper and this grade gave expression to the student's desire to graduate. It is not a common expression (To give expression for hope) but I have heard it.