<p>A recently developed fractional derivative operator involving a generalized exponential kernel, together with its extension to the singular kernel case and the corresponding fractional integral operator, have been systematically formulated and analyzed. This paper aims to introduce a higher-order fractional derivative operator associated with the considered generalized exponential kernel, along with its corresponding fractional integral operator. Specifically, it provides a modified extension of the higher-order fractional derivative operator that incorporates a singular kernel and applies to a broader function space. Several properties and relationships of the proposed operators are derived. As an application, the paper presents numerical simulations of the fractional Helmholtz-Duffing equation involving the generalized exponential kernel derivatives. The extended higher-order operator, which addresses initialization issues in the relevant fractional differential equations, offers potential for further exploration and application within the framework of fractional calculus.</p>