We develop a stochastic integration theory with respect to volatility modulated Volterra processes driven by time-changed Lévy processes. We extend recent results that used Volterra processes driven by either a Brownian motion or a square-integrable, zero-mean pure-jump Lévy processes with stochastic modulation of the amplitude of the volatility. Our approach allows for more general driving noises like, for example, $\alpha$-stable processes. From the modelling perspective, we allow for independent modulation of the amplitude and the intensity of the stochastic volatility obtaining a more flexible modelling framework. This is joint work with Ole Barndorff-Nielsen and Fred Espen Benth.