Online-Offline Iterative Learning Control provides an effective and robust solution to learn precise trajectory tracking when dealing with repetitive tasks. Yet, these algorithms were developed under the assumption that the relative degree between input and output is one. This prevents applications in many practically meaningful situations - e.g. mechanical systems control. To overcome this issue, this manuscript proposes a PI^σ - PI^σ algorithm fusing information from the whole visible dynamics. We provide sufficient convergence conditions when the controlled system has a generic constant relative degree, and it is possibly subject to measurement delay. The controller is validated on several simulation scenarios, including learning to swing-up a soft pendulum.