<p>A general class of fractional integral and derivative operators, which was presented by the mathematical study of fractional calculus in relation to functions, has been usefully used in various areas of applied sciences. In addition, a general framework for Mittag-Leffler type fractional derivatives with respect to another function has been introduced. This framework of fractional calculus involves a composite kernel of the Mittag-Leffler function and an increasing function, which is related to the concept of fractional calculus with respect to functions. In this paper, we describe the general framework for Mittag-Leffler type fractional derivatives as convergent series in terms of generalized fractional integral operators and introduce the associated fractional integral operators. Then, mainly, we extend the general framework of Mittag-Leffler fractional derivatives with respect to functions such that the new extension has an integrable singular kernel. The extended version is able to provide useful insights regarding the modeling issue so that it overcomes the initialization problem of non-singular kernel fractional derivatives. We use a generalization of the Laplace transform to demonstrate some fundamental properties of the studied operators. Finally, to draw attention to the features of the extended version, we present the dynamics of a fractional version of the Lotka-Volterra system that includes the considered extended derivatives.</p>