This research introduced two new contraction mappings by integrating $\psi$-modified interpolative Ćirić-Reich-Rus-type and $F$-contractions with the recently established $P$-contraction framework. Specifically, the concepts of $\psi$-modified interpolative Ćirić-Reich-Rus- $P$-type and $F$- $P$-type contractions were formulated. Existence and uniqueness theorems for fixed points regarding these mappings were established within the context of complete $b$-metric spaces. To substantiate the theoretical findings, several illustrative examples were provided. Furthermore, the practical utility of the results was demonstrated by addressing the existence and uniqueness of solutions for both a specific class of nonlinear Fredholm integral equations and nonlinear fractional differential equations.