Approximating Min-Sum Set Multicover
Yes, this has been studied. It was called the multiple intents re-ranking problem by Azar, Gamzu, and Yin who gave a $\log n$ approximation using a cleverly modified greedy algorithm (the point is being greedy with respect to a clever metric). Bansal, Gupta, and Krishnaswamy had the good sense to rename the problem to generalized min-sum set cover and gave a constant approximation via LP rounding. The constant is very large, more than 400. Skutella and Williamson improved this to 28.

BTW you state the problem as finding an ordering on the sets to satisfy a coverage requirement on the elements. Some of the above papers state the problem in the equivalent dual form: find an ordering on the elements to satisfy hitting requirements on the sets.