<p>A recent singular kernel fractional derivative operator is developed as a proper inverse of a convolution-type fractional integral operator merged with a Prabhakar kernel. In this paper, we introduce higher-order singular kernel fractional derivatives by extending the idea of forming the newly developed fractional derivative, which has similar advantages to the Caputo derivative, and then introduce the associated fractional integrals. We study the unique existence of solutions to IVPs involving the proposed higher-order derivatives. Then we, mainly, formulate a numerical algorithm to deal with IVPs incorporating the studied derivative. We present a comparative study between some noteworthy relationships and features of the studied derivative and Caputo’s derivative. Next, we provide numerical solutions for some models that include the studied derivative, and we conduct numerical comparisons between the solutions in cases of using the studied derivative and the Caputo derivative. The proposed fractional derivative, the presented relationships, and the developed numerical algorithm are expected to subsequently be exploited for further use and application in fractional calculus.</p>