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A: Dad, I got a really cut on my finger from the door handle. It's bleeding a lot!

B: Let me have a look.
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wangqh2696122A: Dad, I got a really cut on my finger from the door handle. It's bleeding a lot!

B: Let me have a look.

B is okay, but A: Dad, I really cut my finger on the door handle.

Really is an adverb and "a cut" is a noun. In your example, you say "a really cut," which doens't work.

Cut is also a verb: I really cut myself = I cut myself severely.

Comments  
Students: Are you brave enough to let our tutors analyse your pronunciation?
What about this response: Don't worry!
It could be a proper response. It depends on the situation. Grammatically, it's okay.