The paper investigates a new type of contraction called -HR-Ćirić-Reich-Rus contractions, which extends the existing research on Ćirić-Reich-Rus contraction mappings. We prove fixed point theorems by enriching -HR-Ćirić-Reich-Rus contractions with respect to the asymptotic regularity of generalized averaged mappings. These mappings are denoted as , where is the identity map and is a function from a normed space to real numbers.