Starting from the numerical solution of the 6-band textbf{k.p} description of a lattice-mismatched ellipsoidal quantum dot situated inside a nanowire, including a spin Zeeman effect with values appropriate to a dilute magnetic semiconductor, we propose and test phenomenological models of the effect of the built-in strain on the heavy hole, light hole and exciton states. We test the validity and the limits of a description restricted to a ($Gamma_8$) quadruplet of ground states and we demonstrate the role of the interactions of the light-hole state with light-hole excited states. We show that the built-in axial strain not only defines the character, heavy-hole or light-hole, of the ground state, but also mixes significantly the light-hole state with the split-off bands states: Even for a spin-orbit energy as large as 1 eV, that mixing induces first-order modifications of properties such as the spin value and anisotropy, the oscillator strength, and the electron-hole exchange, for which we extend the description to the light-hole exciton. CdTe/ZnTe quantum dots are mainly used as a test case but the concepts we discuss apply to many heterostructures, from mismatched II-VI and III-V quantum dots and nanowires, to III-V nanostructures submitted to an applied stress and to silicon nanodevices with even smaller residual strains.
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