However, you say that your native language is English. As far as I know, most English dialects recognise a difference between the sounds you write about.
Cut: in southern BrE and General AmE: /kʌt/; in some northern BrE dialects: /kʊt/
Cat: in most dialects: /kæt/
We are talking about two clearly different vowels when we consider cut and car.
Student or Learner